
@misc{zou_uncertainty_2023,
	title = {Uncertainty quantification for noisy inputs-outputs in physics-informed neural networks and neural operators},
	url = {http://arxiv.org/abs/2311.11262},
	abstract = {Uncertainty quantification (UQ) in scientific machine learning (SciML) becomes increasingly critical as neural networks (NNs) are being widely adopted in addressing complex problems across various scientific disciplines. Representative SciML models are physics-informed neural networks (PINNs) and neural operators (NOs). While UQ in SciML has been increasingly investigated in recent years, very few works have focused on addressing the uncertainty caused by the noisy inputs, such as spatial-temporal coordinates in PINNs and input functions in NOs. The presence of noise in the inputs of the models can pose significantly more challenges compared to noise in the outputs of the models, primarily due to the inherent nonlinearity of most SciML algorithms. As a result, UQ for noisy inputs becomes a crucial factor for reliable and trustworthy deployment of these models in applications involving physical knowledge. To this end, we introduce a Bayesian approach to quantify uncertainty arising from noisy inputs-outputs in PINNs and NOs. We show that this approach can be seamlessly integrated into PINNs and NOs, when they are employed to encode the physical information. PINNs incorporate physics by including physics-informed terms via automatic differentiation, either in the loss function or the likelihood, and often take as input the spatial-temporal coordinate. Therefore, the present method equips PINNs with the capability to address problems where the observed coordinate is subject to noise. On the other hand, pretrained NOs are also commonly employed as equation-free surrogates in solving differential equations and Bayesian inverse problems, in which they take functions as inputs. The proposed approach enables them to handle noisy measurements for both input and output functions with UQ.},
	urldate = {2023-12-17},
	publisher = {arXiv},
	author = {Zou, Zongren and Meng, Xuhui and Karniadakis, George Em},
	month = nov,
	year = {2023},
	note = {arXiv:2311.11262 [physics]},
	keywords = {Computer Science - Machine Learning, Physics - Computational Physics, unread},
}

@misc{krishnapriyan_characterizing_2021,
	title = {Characterizing possible failure modes in physics-informed neural networks},
	url = {http://arxiv.org/abs/2109.01050},
	doi = {10.48550/arXiv.2109.01050},
	abstract = {Recent work in scientific machine learning has developed so-called physics-informed neural network (PINN) models. The typical approach is to incorporate physical domain knowledge as soft constraints on an empirical loss function and use existing machine learning methodologies to train the model. We demonstrate that, while existing PINN methodologies can learn good models for relatively trivial problems, they can easily fail to learn relevant physical phenomena for even slightly more complex problems. In particular, we analyze several distinct situations of widespread physical interest, including learning differential equations with convection, reaction, and diffusion operators. We provide evidence that the soft regularization in PINNs, which involves PDE-based differential operators, can introduce a number of subtle problems, including making the problem more ill-conditioned. Importantly, we show that these possible failure modes are not due to the lack of expressivity in the NN architecture, but that the PINN's setup makes the loss landscape very hard to optimize. We then describe two promising solutions to address these failure modes. The first approach is to use curriculum regularization, where the PINN's loss term starts from a simple PDE regularization, and becomes progressively more complex as the NN gets trained. The second approach is to pose the problem as a sequence-to-sequence learning task, rather than learning to predict the entire space-time at once. Extensive testing shows that we can achieve up to 1-2 orders of magnitude lower error with these methods as compared to regular PINN training.},
	urldate = {2025-01-29},
	publisher = {arXiv},
	author = {Krishnapriyan, Aditi S. and Gholami, Amir and Zhe, Shandian and Kirby, Robert M. and Mahoney, Michael W.},
	month = nov,
	year = {2021},
	note = {arXiv:2109.01050 [cs]},
	keywords = {Computer Science - Artificial Intelligence, Computer Science - Machine Learning, Computer Science - Numerical Analysis, Mathematics - Numerical Analysis, Physics - Computational Physics},
}

@article{penwarden_unified_2023,
	title = {A unified scalable framework for causal sweeping strategies for {Physics}-{Informed} {Neural} {Networks} ({PINNs}) and their temporal decompositions},
	volume = {493},
	issn = {00219991},
	url = {http://arxiv.org/abs/2302.14227},
	doi = {10.1016/j.jcp.2023.112464},
	abstract = {Physics-informed neural networks (PINNs) as a means of solving partial differential equations (PDE) have garnered much attention in the Computational Science and Engineering (CS\&E) world. However, a recent topic of interest is exploring various training (i.e., optimization) challenges - in particular, arriving at poor local minima in the optimization landscape results in a PINN approximation giving an inferior, and sometimes trivial, solution when solving forward time-dependent PDEs with no data. This problem is also found in, and in some sense more difficult, with domain decomposition strategies such as temporal decomposition using XPINNs. We furnish examples and explanations for different training challenges, their cause, and how they relate to information propagation and temporal decomposition. We then propose a new stacked-decomposition method that bridges the gap between time-marching PINNs and XPINNs. We also introduce significant computational speed-ups by using transfer learning concepts to initialize subnetworks in the domain and loss tolerance-based propagation for the subdomains. Finally, we formulate a new time-sweeping collocation point algorithm inspired by the previous PINNs causality literature, which our framework can still describe, and provides a significant computational speed-up via reduced-cost collocation point segmentation. The proposed methods form our unified framework, which overcomes training challenges in PINNs and XPINNs for time-dependent PDEs by respecting the causality in multiple forms and improving scalability by limiting the computation required per optimization iteration. Finally, we provide numerical results for these methods on baseline PDE problems for which unmodified PINNs and XPINNs struggle to train.},
	urldate = {2025-01-29},
	journal = {Journal of Computational Physics},
	author = {Penwarden, Michael and Jagtap, Ameya D. and Zhe, Shandian and Karniadakis, George Em and Kirby, Robert M.},
	month = nov,
	year = {2023},
	note = {arXiv:2302.14227 [physics]},
	keywords = {Computer Science - Machine Learning, Physics - Computational Physics, unread},
	pages = {112464},
}

@misc{wang_respecting_2022,
	title = {Respecting causality is all you need for training physics-informed neural networks},
	url = {http://arxiv.org/abs/2203.07404},
	doi = {10.48550/arXiv.2203.07404},
	abstract = {While the popularity of physics-informed neural networks (PINNs) is steadily rising, to this date PINNs have not been successful in simulating dynamical systems whose solution exhibits multi-scale, chaotic or turbulent behavior. In this work we attribute this shortcoming to the inability of existing PINNs formulations to respect the spatio-temporal causal structure that is inherent to the evolution of physical systems. We argue that this is a fundamental limitation and a key source of error that can ultimately steer PINN models to converge towards erroneous solutions. We address this pathology by proposing a simple re-formulation of PINNs loss functions that can explicitly account for physical causality during model training. We demonstrate that this simple modification alone is enough to introduce significant accuracy improvements, as well as a practical quantitative mechanism for assessing the convergence of a PINNs model. We provide state-of-the-art numerical results across a series of benchmarks for which existing PINNs formulations fail, including the chaotic Lorenz system, the Kuramoto-Sivashinsky equation in the chaotic regime, and the Navier-Stokes equations in the turbulent regime. To the best of our knowledge, this is the first time that PINNs have been successful in simulating such systems, introducing new opportunities for their applicability to problems of industrial complexity.},
	urldate = {2025-01-29},
	publisher = {arXiv},
	author = {Wang, Sifan and Sankaran, Shyam and Perdikaris, Paris},
	month = mar,
	year = {2022},
	note = {arXiv:2203.07404 [cs]},
	keywords = {Computer Science - Machine Learning, Computer Science - Numerical Analysis, Mathematics - Numerical Analysis, Nonlinear Sciences - Chaotic Dynamics, Physics - Fluid Dynamics, Statistics - Machine Learning, unread},
}

@misc{bingham_pyro_2018,
	title = {Pyro: {Deep} {Universal} {Probabilistic} {Programming}},
	shorttitle = {Pyro},
	url = {http://arxiv.org/abs/1810.09538},
	doi = {10.48550/arXiv.1810.09538},
	abstract = {Pyro is a probabilistic programming language built on Python as a platform for developing advanced probabilistic models in AI research. To scale to large datasets and high-dimensional models, Pyro uses stochastic variational inference algorithms and probability distributions built on top of PyTorch, a modern GPU-accelerated deep learning framework. To accommodate complex or model-specific algorithmic behavior, Pyro leverages Poutine, a library of composable building blocks for modifying the behavior of probabilistic programs.},
	urldate = {2024-10-11},
	publisher = {arXiv},
	author = {Bingham, Eli and Chen, Jonathan P. and Jankowiak, Martin and Obermeyer, Fritz and Pradhan, Neeraj and Karaletsos, Theofanis and Singh, Rohit and Szerlip, Paul and Horsfall, Paul and Goodman, Noah D.},
	month = oct,
	year = {2018},
	note = {arXiv:1810.09538 [cs, stat]},
	keywords = {Computer Science - Machine Learning, Computer Science - Programming Languages, Statistics - Machine Learning, unread},
}

@inproceedings{noauthor_notitle_nodate,
	keywords = {unread},
}

@article{akrami_landscape_2019,
	title = {The landscape, the swampland and the era of precision cosmology},
	volume = {67},
	issn = {0015-8208, 1521-3978},
	url = {http://arxiv.org/abs/1808.09440},
	doi = {10.1002/prop.201800075},
	abstract = {We review the advanced version of the KKLT construction and pure \$d=4\$ de Sitter supergravity, involving a nilpotent multiplet, with regard to various conjectures that de Sitter state cannot exist in string theory. We explain why we consider these conjectures problematic and not well motivated, and why the recently proposed alternative string theory models of dark energy, ignoring vacuum stabilization, are ruled out by cosmological observations at least at the \$3{\textbackslash}sigma\$ level, i.e. with more than \$99.7{\textbackslash}\%\$ confidence.},
	number = {1-2},
	urldate = {2024-10-09},
	journal = {Fortschritte der Physik},
	author = {Akrami, Yashar and Kallosh, Renata and Linde, Andrei and Vardanyan, Valeri},
	month = jan,
	year = {2019},
	note = {arXiv:1808.09440 [astro-ph, physics:gr-qc, physics:hep-ph, physics:hep-th, physics:math-ph]},
	keywords = {Astrophysics - Cosmology and Nongalactic Astrophysics, General Relativity and Quantum Cosmology, High Energy Physics - Phenomenology, High Energy Physics - Theory, Mathematical Physics, unread},
	pages = {1800075},
}

@misc{motta_taxonomy_2021,
	title = {Taxonomy of {Dark} {Energy} {Models}},
	url = {http://arxiv.org/abs/2104.04642},
	doi = {10.48550/arXiv.2104.04642},
	abstract = {The accelerated expansion of the Universe is one of the main discoveries of the past decades, indicating the presence of an unknown component: the dark energy. Evidence of its presence is being gathered by a succession of observational experiments with increasing precision in its measurements. However, the most accepted model for explaining the dynamic of our Universe, the so-called Lambda cold dark matter, face several problems related to the nature of such energy component. This has lead to a growing exploration of alternative models attempting to solve those drawbacks. In this review, we briefly summarize the characteristics of a (non-exhaustive) list of dark energy models as well as some of the most used cosmological samples. Next, we discuss how to constrain each model's parameters using observational data. Finally, we summarize the status of dark energy modeling.},
	urldate = {2024-10-09},
	publisher = {arXiv},
	author = {Motta, V. and García-Aspeitia, Miguel A. and Hernández-Almada, A. and Magaña, J. and Verdugo, Tomás},
	month = apr,
	year = {2021},
	note = {arXiv:2104.04642 [astro-ph, physics:gr-qc]},
	keywords = {Astrophysics - Cosmology and Nongalactic Astrophysics, General Relativity and Quantum Cosmology, unread},
}

@article{dagostino_measurements_2020,
	title = {Measurements of \${H}\_0\$ in modified gravity theories: {The} role of lensed quasars in the late-time {Universe}},
	volume = {101},
	issn = {2470-0010, 2470-0029},
	shorttitle = {Measurements of \${H}\_0\$ in modified gravity theories},
	url = {http://arxiv.org/abs/2002.06381},
	doi = {10.1103/PhysRevD.101.103505},
	abstract = {In this work, we obtain measurements of the Hubble constant in the context of modified gravity theories. We set up our theoretical framework by considering viable cosmological \$f(R)\$ and \$f(T)\$ models, and we analyzed them through the use of geometrical data sets obtained in a model-independent way, namely, gravitationally lensed quasars with measured time delays, standard clocks from cosmic chronometers, and standard candles from the Pantheon Supernovae Ia sample. We find \$H\_0=(72.4{\textbackslash}pm 1.4)\$ km s\${\textasciicircum}\{-1\}\$ Mpc\${\textasciicircum}\{-1\}\$ and \$H\_0=(71.5{\textbackslash}pm 1.3)\$ km s\${\textasciicircum}\{-1\}\$ Mpc\${\textasciicircum}\{-1\}\$ for the \$f(R)\$ and \$f(T)\$ models, respectively. Our results represent 1.9{\textbackslash}\% and 1.8{\textbackslash}\% measurements of the Hubble constant, which are fully consistent with the local estimate of \$H\_0\$ by the Hubble Space Telescope. We do not find significant departures from general relativity, as our study shows that the characteristic parameters of the extensions of gravity beyond general relativity are compatible with the \${\textbackslash}Lambda\$CDM cosmology. Moreover, within the standard cosmological framework, our full joint analysis suggests that it is possible to measure the dark energy equation of state parameter at 1.2{\textbackslash}\% accuracy, although we find no statistical evidence for deviations from the cosmological constant case.},
	number = {10},
	urldate = {2024-10-09},
	journal = {Physical Review D},
	author = {D'Agostino, Rocco and Nunes, Rafael C.},
	month = may,
	year = {2020},
	note = {arXiv:2002.06381 [astro-ph, physics:gr-qc, physics:hep-ph]},
	keywords = {Astrophysics - Cosmology and Nongalactic Astrophysics, General Relativity and Quantum Cosmology, High Energy Physics - Phenomenology, unread},
	pages = {103505},
}

@article{meng_learning_2022,
	title = {Learning {Functional} {Priors} and {Posteriors} from {Data} and {Physics}},
	volume = {457},
	issn = {00219991},
	url = {http://arxiv.org/abs/2106.05863},
	doi = {10.1016/j.jcp.2022.111073},
	abstract = {We develop a new Bayesian framework based on deep neural networks to be able to extrapolate in space-time using historical data and to quantify uncertainties arising from both noisy and gappy data in physical problems. Specifically, the proposed approach has two stages: (1) prior learning and (2) posterior estimation. At the first stage, we employ the physics-informed Generative Adversarial Networks (PI-GAN) to learn a functional prior either from a prescribed function distribution, e.g., Gaussian process, or from historical data and physics. At the second stage, we employ the Hamiltonian Monte Carlo (HMC) method to estimate the posterior in the latent space of PI-GANs. In addition, we use two different approaches to encode the physics: (1) automatic differentiation, used in the physics-informed neural networks (PINNs) for scenarios with explicitly known partial differential equations (PDEs), and (2) operator regression using the deep operator network (DeepONet) for PDE-agnostic scenarios. We then test the proposed method for (1) meta-learning for one-dimensional regression, and forward/inverse PDE problems (combined with PINNs); (2) PDE-agnostic physical problems (combined with DeepONet), e.g., fractional diffusion as well as saturated stochastic (100-dimensional) flows in heterogeneous porous media; and (3) spatial-temporal regression problems, i.e., inference of a marine riser displacement field. The results demonstrate that the proposed approach can provide accurate predictions as well as uncertainty quantification given very limited scattered and noisy data, since historical data could be available to provide informative priors. In summary, the proposed method is capable of learning flexible functional priors, and can be extended to big data problems using stochastic HMC or normalizing flows since the latent space is generally characterized as low dimensional.},
	urldate = {2024-09-10},
	journal = {Journal of Computational Physics},
	author = {Meng, Xuhui and Yang, Liu and Mao, Zhiping and Ferrandis, Jose del Aguila and Karniadakis, George Em},
	month = may,
	year = {2022},
	note = {arXiv:2106.05863 [cs]},
	keywords = {Computer Science - Machine Learning, unread},
	pages = {111073},
}

@book{astrom_feedback_2008,
	address = {Princeton, NJ},
	title = {Feedback systems: an introduction for scientists and engineers},
	isbn = {978-0-691-13576-2},
	shorttitle = {Feedback systems},
	language = {en},
	publisher = {Princeton Univ. Press},
	author = {Åström, Karl J. and Murray, Richard M.},
	year = {2008},
	keywords = {unread},
}

@book{astrom_feedback_2008-1,
	address = {Princeton, NJ},
	title = {Feedback systems},
	language = {en},
	publisher = {Princeton University Press},
	author = {Astrom, Karl Johan and Murray, Richard M},
	month = apr,
	year = {2008},
	keywords = {unread},
}

@article{brunton_discovering_2016,
	title = {Discovering governing equations from data: {Sparse} identification of nonlinear dynamical systems},
	volume = {113},
	issn = {0027-8424, 1091-6490},
	shorttitle = {Discovering governing equations from data},
	url = {http://arxiv.org/abs/1509.03580},
	doi = {10.1073/pnas.1517384113},
	abstract = {The ability to discover physical laws and governing equations from data is one of humankind's greatest intellectual achievements. A quantitative understanding of dynamic constraints and balances in nature has facilitated rapid development of knowledge and enabled advanced technological achievements, including aircraft, combustion engines, satellites, and electrical power. In this work, we combine sparsity-promoting techniques and machine learning with nonlinear dynamical systems to discover governing physical equations from measurement data. The only assumption about the structure of the model is that there are only a few important terms that govern the dynamics, so that the equations are sparse in the space of possible functions; this assumption holds for many physical systems. In particular, we use sparse regression to determine the fewest terms in the dynamic governing equations required to accurately represent the data. The resulting models are parsimonious, balancing model complexity with descriptive ability while avoiding overfitting. We demonstrate the algorithm on a wide range of problems, from simple canonical systems, including linear and nonlinear oscillators and the chaotic Lorenz system, to the fluid vortex shedding behind an obstacle. The fluid example illustrates the ability of this method to discover the underlying dynamics of a system that took experts in the community nearly 30 years to resolve. We also show that this method generalizes to parameterized, time-varying, or externally forced systems.},
	number = {15},
	urldate = {2024-07-05},
	journal = {Proceedings of the National Academy of Sciences},
	author = {Brunton, Steven L. and Proctor, Joshua L. and Kutz, J. Nathan},
	month = apr,
	year = {2016},
	note = {arXiv:1509.03580 [math]},
	keywords = {Mathematics - Dynamical Systems, unread},
	pages = {3932--3937},
}

@book{boyce_elementary_2008,
	address = {Chichester, England},
	edition = {9},
	title = {Elementary differential equations and boundary value problems},
	language = {en},
	publisher = {John Wiley \& Sons},
	author = {Boyce, William E and DiPrima, Richard C},
	month = oct,
	year = {2008},
	keywords = {unread},
}

@article{jimenez_constraining_2002,
	title = {Constraining {Cosmological} {Parameters} {Based} on {Relative} {Galaxy} {Ages}},
	volume = {573},
	issn = {0004-637X},
	url = {https://iopscience.iop.org/article/10.1086/340549/meta},
	doi = {10.1086/340549},
	abstract = {Constraining Cosmological Parameters Based on Relative Galaxy Ages, Raul Jimenez, Abraham Loeb},
	language = {en},
	number = {1},
	urldate = {2024-04-21},
	journal = {The Astrophysical Journal},
	author = {Jimenez, Raul and Loeb, Abraham},
	month = jul,
	year = {2002},
	note = {Publisher: IOP Publishing},
	keywords = {unread},
	pages = {37},
}

@article{hu_models_2007,
	title = {Models of \$f({R})\$ cosmic acceleration that evade solar system tests},
	volume = {76},
	url = {https://link.aps.org/doi/10.1103/PhysRevD.76.064004},
	doi = {10.1103/PhysRevD.76.064004},
	abstract = {We study a class of metric-variation f(R) models that accelerates the expansion without a cosmological constant and satisfies both cosmological and solar-system tests in the small-field limit of the parameter space. Solar-system tests alone place only weak bounds on these models, since the additional scalar degree of freedom is locked to the high-curvature general-relativistic prediction across more than 25 orders of magnitude in density, out through the solar corona. This agreement requires that the galactic halo be of sufficient extent to maintain the galaxy at high curvature in the presence of the low-curvature cosmological background. If the galactic halo and local environment in f(R) models do not have substantially deeper potentials than expected in ΛCDM, then cosmological field amplitudes {\textbar}fR{\textbar}≳10−6 will cause the galactic interior to evolve to low curvature during the acceleration epoch. Viability of large-deviation models therefore rests on the structure and evolution of the galactic halo, requiring cosmological simulations of f(R) models, and not directly on solar-system tests. Even small deviations that conservatively satisfy both galactic and solar-system constraints can still be tested by future, percent-level measurements of the linear power spectrum, while they remain undetectable to cosmological-distance measures. Although we illustrate these effects in a specific class of models, the requirements on f(R) are phrased in a nearly model-independent manner.},
	number = {6},
	urldate = {2024-04-20},
	journal = {Physical Review D},
	author = {Hu, Wayne and Sawicki, Ignacy},
	month = sep,
	year = {2007},
	note = {Publisher: American Physical Society},
	keywords = {unread},
	pages = {064004},
}

@misc{paszke_pytorch_2019,
	title = {{PyTorch}: {An} {Imperative} {Style}, {High}-{Performance} {Deep} {Learning} {Library}},
	shorttitle = {{PyTorch}},
	url = {http://arxiv.org/abs/1912.01703},
	doi = {10.48550/arXiv.1912.01703},
	abstract = {Deep learning frameworks have often focused on either usability or speed, but not both. PyTorch is a machine learning library that shows that these two goals are in fact compatible: it provides an imperative and Pythonic programming style that supports code as a model, makes debugging easy and is consistent with other popular scientific computing libraries, while remaining efficient and supporting hardware accelerators such as GPUs. In this paper, we detail the principles that drove the implementation of PyTorch and how they are reflected in its architecture. We emphasize that every aspect of PyTorch is a regular Python program under the full control of its user. We also explain how the careful and pragmatic implementation of the key components of its runtime enables them to work together to achieve compelling performance. We demonstrate the efficiency of individual subsystems, as well as the overall speed of PyTorch on several common benchmarks.},
	urldate = {2024-02-09},
	publisher = {arXiv},
	author = {Paszke, Adam and Gross, Sam and Massa, Francisco and Lerer, Adam and Bradbury, James and Chanan, Gregory and Killeen, Trevor and Lin, Zeming and Gimelshein, Natalia and Antiga, Luca and Desmaison, Alban and Köpf, Andreas and Yang, Edward and DeVito, Zach and Raison, Martin and Tejani, Alykhan and Chilamkurthy, Sasank and Steiner, Benoit and Fang, Lu and Bai, Junjie and Chintala, Soumith},
	month = dec,
	year = {2019},
	note = {arXiv:1912.01703 [cs, stat]},
	keywords = {Computer Science - Machine Learning, Computer Science - Mathematical Software, Statistics - Machine Learning, unread},
}

@misc{paszke_pytorch_2019-1,
	title = {{PyTorch}: {An} {Imperative} {Style}, {High}-{Performance} {Deep} {Learning} {Library}},
	shorttitle = {{PyTorch}},
	url = {http://arxiv.org/abs/1912.01703},
	doi = {10.48550/arXiv.1912.01703},
	abstract = {Deep learning frameworks have often focused on either usability or speed, but not both. PyTorch is a machine learning library that shows that these two goals are in fact compatible: it provides an imperative and Pythonic programming style that supports code as a model, makes debugging easy and is consistent with other popular scientific computing libraries, while remaining efficient and supporting hardware accelerators such as GPUs. In this paper, we detail the principles that drove the implementation of PyTorch and how they are reflected in its architecture. We emphasize that every aspect of PyTorch is a regular Python program under the full control of its user. We also explain how the careful and pragmatic implementation of the key components of its runtime enables them to work together to achieve compelling performance. We demonstrate the efficiency of individual subsystems, as well as the overall speed of PyTorch on several common benchmarks.},
	urldate = {2024-02-09},
	publisher = {arXiv},
	author = {Paszke, Adam and Gross, Sam and Massa, Francisco and Lerer, Adam and Bradbury, James and Chanan, Gregory and Killeen, Trevor and Lin, Zeming and Gimelshein, Natalia and Antiga, Luca and Desmaison, Alban and Köpf, Andreas and Yang, Edward and DeVito, Zach and Raison, Martin and Tejani, Alykhan and Chilamkurthy, Sasank and Steiner, Benoit and Fang, Lu and Bai, Junjie and Chintala, Soumith},
	month = dec,
	year = {2019},
	note = {arXiv:1912.01703 [cs, stat]},
	keywords = {Computer Science - Machine Learning, Computer Science - Mathematical Software, Statistics - Machine Learning, unread},
}

@article{raissi_physics-informed_2019,
	title = {Physics-informed neural networks: {A} deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations},
	volume = {378},
	issn = {0021-9991},
	shorttitle = {Physics-informed neural networks},
	url = {https://www.sciencedirect.com/science/article/pii/S0021999118307125},
	doi = {10.1016/j.jcp.2018.10.045},
	abstract = {We introduce physics-informed neural networks – neural networks that are trained to solve supervised learning tasks while respecting any given laws of physics described by general nonlinear partial differential equations. In this work, we present our developments in the context of solving two main classes of problems: data-driven solution and data-driven discovery of partial differential equations. Depending on the nature and arrangement of the available data, we devise two distinct types of algorithms, namely continuous time and discrete time models. The first type of models forms a new family of data-efficient spatio-temporal function approximators, while the latter type allows the use of arbitrarily accurate implicit Runge–Kutta time stepping schemes with unlimited number of stages. The effectiveness of the proposed framework is demonstrated through a collection of classical problems in fluids, quantum mechanics, reaction–diffusion systems, and the propagation of nonlinear shallow-water waves.},
	urldate = {2024-02-08},
	journal = {Journal of Computational Physics},
	author = {Raissi, M. and Perdikaris, P. and Karniadakis, G. E.},
	month = feb,
	year = {2019},
	keywords = {Data-driven scientific computing, Machine learning, Nonlinear dynamics, Predictive modeling, Runge–Kutta methods},
	pages = {686--707},
}

@article{roy_comprehensive_2011,
	title = {A comprehensive framework for verification, validation, and uncertainty quantification in scientific computing},
	volume = {200},
	issn = {00457825},
	url = {https://linkinghub.elsevier.com/retrieve/pii/S0045782511001290},
	doi = {10.1016/j.cma.2011.03.016},
	abstract = {An overview of a comprehensive framework is given for estimating the predictive uncertainty of scientiﬁc computing applications. The framework is comprehensive in the sense that it treats both types of uncertainty (aleatory and epistemic), incorporates uncertainty due to the mathematical form of the model, and it provides a procedure for including estimates of numerical error in the predictive uncertainty. Aleatory (random) uncertainties in model inputs are treated as random variables, while epistemic (lack of knowledge) uncertainties are treated as intervals with no assumed probability distributions. Approaches for propagating both types of uncertainties through the model to the system response quantities of interest are brieﬂy discussed. Numerical approximation errors (due to discretization, iteration, and computer round off) are estimated using veriﬁcation techniques, and the conversion of these errors into epistemic uncertainties is discussed. Model form uncertainty is quantiﬁed using (a) model validation procedures, i.e., statistical comparisons of model predictions to available experimental data, and (b) extrapolation of this uncertainty structure to points in the application domain where experimental data do not exist. Finally, methods for conveying the total predictive uncertainty to decision makers are presented. The different steps in the predictive uncertainty framework are illustrated using a simple example in computational ﬂuid dynamics applied to a hypersonic wind tunnel.},
	language = {en},
	number = {25-28},
	urldate = {2023-10-11},
	journal = {Computer Methods in Applied Mechanics and Engineering},
	author = {Roy, Christopher J. and Oberkampf, William L.},
	month = jun,
	year = {2011},
	pages = {2131--2144},
}

@article{psaros_uncertainty_2023,
	title = {Uncertainty quantification in scientific machine learning: {Methods}, metrics, and comparisons},
	volume = {477},
	issn = {0021-9991},
	shorttitle = {Uncertainty quantification in scientific machine learning},
	url = {https://www.sciencedirect.com/science/article/pii/S0021999122009652},
	doi = {10.1016/j.jcp.2022.111902},
	abstract = {Neural networks (NNs) are currently changing the computational paradigm on how to combine data with mathematical laws in physics and engineering in a profound way, tackling challenging inverse and ill-posed problems not solvable with traditional methods. However, quantifying errors and uncertainties in NN-based inference is more complicated than in traditional methods. This is because in addition to aleatoric uncertainty associated with noisy data, there is also uncertainty due to limited data, but also due to NN hyperparameters, overparametrization, optimization and sampling errors as well as model misspecification. Although there are some recent works on uncertainty quantification (UQ) in NNs, there is no systematic investigation of suitable methods towards quantifying the total uncertainty effectively and efficiently even for function approximation, and there is even less work on solving partial differential equations and learning operator mappings between infinite-dimensional function spaces using NNs. In this work, we present a comprehensive framework that includes uncertainty modeling, new and existing solution methods, as well as evaluation metrics and post-hoc improvement approaches. To demonstrate the applicability and reliability of our framework, we present an extensive comparative study in which various methods are tested on prototype problems, including problems with mixed input-output data, and stochastic problems in high dimensions. In the Appendix, we include a comprehensive description of all the UQ methods employed. Further, to help facilitate the deployment of UQ in Scientific Machine Learning research and practice, we present and develop in [1] an open-source Python library (github.com/Crunch-UQ4MI/neuraluq), termed NeuralUQ, that is accompanied by an educational tutorial and additional computational experiments.},
	urldate = {2023-10-11},
	journal = {Journal of Computational Physics},
	author = {Psaros, Apostolos F. and Meng, Xuhui and Zou, Zongren and Guo, Ling and Karniadakis, George Em},
	month = mar,
	year = {2023},
	keywords = {Bayesian framework, Neural operator learning, Physics-informed neural networks, Scientific machine learning, Stochastic partial differential equations, Uncertainty quantification},
	pages = {111902},
}

@article{abdar_review_2021,
	title = {A review of uncertainty quantification in deep learning: {Techniques}, applications and challenges},
	volume = {76},
	issn = {1566-2535},
	shorttitle = {A review of uncertainty quantification in deep learning},
	url = {https://www.sciencedirect.com/science/article/pii/S1566253521001081},
	doi = {10.1016/j.inffus.2021.05.008},
	abstract = {Uncertainty quantification (UQ) methods play a pivotal role in reducing the impact of uncertainties during both optimization and decision making processes. They have been applied to solve a variety of real-world problems in science and engineering. Bayesian approximation and ensemble learning techniques are two widely-used types of uncertainty quantification (UQ) methods. In this regard, researchers have proposed different UQ methods and examined their performance in a variety of applications such as computer vision (e.g., self-driving cars and object detection), image processing (e.g., image restoration), medical image analysis (e.g., medical image classification and segmentation), natural language processing (e.g., text classification, social media texts and recidivism risk-scoring), bioinformatics, etc. This study reviews recent advances in UQ methods used in deep learning, investigates the application of these methods in reinforcement learning, and highlights fundamental research challenges and directions associated with UQ.},
	urldate = {2023-10-11},
	journal = {Information Fusion},
	author = {Abdar, Moloud and Pourpanah, Farhad and Hussain, Sadiq and Rezazadegan, Dana and Liu, Li and Ghavamzadeh, Mohammad and Fieguth, Paul and Cao, Xiaochun and Khosravi, Abbas and Acharya, U. Rajendra and Makarenkov, Vladimir and Nahavandi, Saeid},
	month = dec,
	year = {2021},
	keywords = {Artificial intelligence, Bayesian statistics, Deep learning, Ensemble learning, Machine learning, Uncertainty quantification},
	pages = {243--297},
}

@article{talavera_differential_nodate,
	title = {Differential equations system to describe bacterial growth, {pH} variation and lactic acid production in batch fermentation},
	language = {en},
	author = {Talavera, Eng Emmanuel Rodríguez and Antonio, Dr Paul and Trujillo, Valle and Muñoz, Dra Yolocuauhtli Salazar},
	keywords = {Read, notion, unread},
}

@inproceedings{snoek_scalable_2015,
	title = {Scalable {Bayesian} {Optimization} {Using} {Deep} {Neural} {Networks}},
	url = {https://www.semanticscholar.org/paper/Scalable-Bayesian-Optimization-Using-Deep-Neural-Snoek-Rippel/93bc65d2842b8cc5f3cf72ebc5b8f75daeacea35},
	abstract = {Bayesian optimization is an effective methodology for the global optimization of functions with expensive evaluations. It relies on querying a distribution over functions defined by a relatively cheap surrogate model. An accurate model for this distribution over functions is critical to the effectiveness of the approach, and is typically fit using Gaussian processes (GPs). However, since GPs scale cubically with the number of observations, it has been challenging to handle objectives whose optimization requires many evaluations, and as such, massively parallelizing the optimization. 
 
In this work, we explore the use of neural networks as an alternative to GPs to model distributions over functions. We show that performing adaptive basis function regression with a neural network as the parametric form performs competitively with state-of-the-art GP-based approaches, but scales linearly with the number of data rather than cubically. This allows us to achieve a previously intractable degree of parallelism, which we apply to large scale hyperparameter optimization, rapidly finding competitive models on benchmark object recognition tasks using convolutional networks, and image caption generation using neural language models.},
	urldate = {2023-09-28},
	author = {Snoek, Jasper and Rippel, Oren and Swersky, Kevin and Kiros, Ryan and Satish, N. and Sundaram, N. and Patwary, Md Mostofa Ali and Prabhat and Adams, Ryan P.},
	month = feb,
	year = {2015},
	keywords = {unread},
}

@article{psaros_uncertainty_2022,
	title = {Uncertainty {Quantification} in {Scientific} {Machine} {Learning}: {Methods}, {Metrics}, and {Comparisons}},
	doi = {10.48550/arxiv.2201.07766},
	abstract = {Neural networks (NNs) are currently changing the computational paradigm on
how to combine data with mathematical laws in physics and engineering in a
profound way, tackling challenging inverse and ill-posed problems not solvable
with traditional methods. However, quantifying errors and uncertainties in
NN-based inference is more complicated than in traditional methods. This is
because in addition to aleatoric uncertainty associated with noisy data, there
is also uncertainty due to limited data, but also due to NN hyperparameters,
overparametrization, optimization and sampling errors as well as model
misspecification. Although there are some recent works on uncertainty
quantification (UQ) in NNs, there is no systematic investigation of suitable
methods towards quantifying the total uncertainty effectively and efficiently
even for function approximation, and there is even less work on solving partial
differential equations and learning operator mappings between
infinite-dimensional function spaces using NNs. In this work, we present a
comprehensive framework that includes uncertainty modeling, new and existing
solution methods, as well as evaluation metrics and post-hoc improvement
approaches. To demonstrate the applicability and reliability of our framework,
we present an extensive comparative study in which various methods are tested
on prototype problems, including problems with mixed input-output data, and
stochastic problems in high dimensions. In the Appendix, we include a
comprehensive description of all the UQ methods employed, which we will make
available as open-source library of all codes included in this framework.},
	author = {Psaros, Apostolos F and Meng, Xuhui and Zou, Zongren and Guo, Ling and Karniadakis, George Em},
	year = {2022},
	note = {arXiv: 2201.07766},
	keywords = {DEs, notion, unread},
}

@incollection{neal_monte_1996,
	address = {New York, NY},
	series = {Lecture {Notes} in {Statistics}},
	title = {Monte {Carlo} {Implementation}},
	isbn = {978-1-4612-0745-0},
	url = {https://doi.org/10.1007/978-1-4612-0745-0_3},
	abstract = {This chapter presents a Markov chain Monte Carlo implementation of Bayesian learning for neural networks in which network parameters are updated using the hybrid Monte Carlo algorithm, a form of the Metropolis algorithm in which candidate states are found by means of dynamical simulation. Hyperparameters are updated separately using Gibbs sampling, allowing their values to be used in chosing good stepsizes for the discretized dynamics. I show that hybrid Monte Carlo performs better than simple Metropolis,due to its avoidance of random walk behaviour. I also discuss variants of hybrid Monte Carlo in which dynamical computations are done using “partial gradients”, in which acceptance is based on a “window” of states,and in which momentum updates incorporate “persistence”.},
	language = {en},
	urldate = {2023-09-28},
	booktitle = {Bayesian {Learning} for {Neural} {Networks}},
	publisher = {Springer},
	author = {Neal, Radford M.},
	editor = {Neal, Radford M.},
	year = {1996},
	doi = {10.1007/978-1-4612-0745-0_3},
	keywords = {Bayesian Learning, Gibbs Sampling, Hide Unit, Metropolis Algorithm, Monte Carlo Estimate, unread},
	pages = {55--98},
}

@article{moreno-zambrano_mathematical_2018,
	title = {A mathematical model of cocoa bean fermentation},
	volume = {5},
	issn = {2054-5703},
	url = {https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6227950/},
	doi = {10.1098/rsos.180964},
	abstract = {Cocoa bean fermentation relies on the sequential activation of several microbial populations, triggering a temporal pattern of biochemical transformations. Understanding this complex process is of tremendous importance as it is known to form the precursors of the resulting chocolate’s flavour and taste. At the same time, cocoa bean fermentation is one of the least controlled processes in the food industry. Here, a quantitative model of cocoa bean fermentation is constructed based on available microbiological and biochemical knowledge. The model is formulated as a system of coupled ordinary differential equations with two distinct types of state variables: (i) metabolite concentrations of glucose, fructose, ethanol, lactic acid and acetic acid and (ii) population sizes of yeast, lactic acid bacteria and acetic acid bacteria. We demonstrate that the model can quantitatively describe existing fermentation time series and that the estimated parameters, obtained by a Bayesian framework, can be used to extract and interpret differences in environmental conditions. The proposed model is a valuable tool towards a mechanistic understanding of this complex biochemical process, and can serve as a starting point for hypothesis testing of new systemic adjustments. In addition to providing the first quantitative mathematical model of cocoa bean fermentation, the purpose of our investigation is to show how differences in estimated parameter values for two experiments allow us to deduce differences in experimental conditions.},
	number = {10},
	urldate = {2022-12-22},
	journal = {Royal Society Open Science},
	author = {Moreno-Zambrano, Mauricio and Grimbs, Sergio and Ullrich, Matthias S. and Hütt, Marc-Thorsten},
	month = oct,
	year = {2018},
	pmid = {30473841},
	pmcid = {PMC6227950},
	keywords = {Read, notion, unread},
	pages = {180964},
}

@article{metropolis_equation_2004,
	title = {Equation of {State} {Calculations} by {Fast} {Computing} {Machines}},
	volume = {21},
	issn = {0021-9606},
	url = {https://doi.org/10.1063/1.1699114},
	doi = {10.1063/1.1699114},
	abstract = {A general method, suitable for fast computing machines, for investigating such properties as equations of state for substances consisting of interacting individual molecules is described. The method consists of a modified Monte Carlo integration over configuration space. Results for the two‐dimensional rigid‐sphere system have been obtained on the Los Alamos MANIAC and are presented here. These results are compared to the free volume equation of state and to a four‐term virial coefficient expansion.},
	number = {6},
	urldate = {2023-09-29},
	journal = {The Journal of Chemical Physics},
	author = {Metropolis, Nicholas and Rosenbluth, Arianna W. and Rosenbluth, Marshall N. and Teller, Augusta H. and Teller, Edward},
	month = dec,
	year = {2004},
	keywords = {unread},
	pages = {1087--1092},
}

@misc{liu_evaluating_2022,
	title = {Evaluating {Error} {Bound} for {Physics}-{Informed} {Neural} {Networks} on {Linear} {Dynamical} {Systems}},
	url = {http://arxiv.org/abs/2207.01114},
	doi = {10.48550/arXiv.2207.01114},
	abstract = {There have been extensive studies on solving differential equations using physics-informed neural networks. While this method has proven advantageous in many cases, a major criticism lies in its lack of analytical error bounds. Therefore, it is less credible than its traditional counterparts, such as the finite difference method. This paper shows that one can mathematically derive explicit error bounds for physics-informed neural networks trained on a class of linear systems of differential equations. More importantly, evaluating such error bounds only requires evaluating the differential equation residual infinity norm over the domain of interest. Our work shows a link between network residuals, which is known and used as loss function, and the absolute error of solution, which is generally unknown. Our approach is semi-phenomonological and independent of knowledge of the actual solution or the complexity or architecture of the network. Using the method of manufactured solution on linear ODEs and system of linear ODEs, we empirically verify the error evaluation algorithm and demonstrate that the actual error strictly lies within our derived bound.},
	urldate = {2022-11-29},
	publisher = {arXiv},
	author = {Liu, Shuheng and Huang, Xiyue and Protopapas, Pavlos},
	month = jul,
	year = {2022},
	note = {arXiv:2207.01114 [cs, math]},
	keywords = {Computer Science - Neural and Evolutionary Computing, Mathematics - Numerical Analysis, unread},
}

@book{liu_residual-based_2023,
	title = {Residual-based error bound for physics-informed neural networks},
	abstract = {Neural networks are universal approximators and are studied for their use in solving differential equations. However, a major criticism is the lack of error bounds for obtained solutions. This paper proposes a technique to rigorously evaluate the error bound of Physics-Informed Neural Networks (PINNs) on most linear ordinary differential equations (ODEs), certain nonlinear ODEs, and first-order linear partial differential equations (PDEs). The error bound is based purely on equation structure and residual information and does not depend on assumptions of how well the networks are trained. We propose algorithms that bound the error efficiently. Some proposed algorithms provide tighter bounds than others at the cost of longer run time.},
	author = {Liu, Shuheng and Huang, Xiyue and Protopapas, Pavlos},
	month = jun,
	year = {2023},
	keywords = {unread},
}

@article{lagaris_artificial_1997,
	title = {Artificial {Neural} {Networks} for {Solving} {Ordinary} and {Partial} {Differential} {Equations}},
	volume = {9},
	url = {http://arxiv.org/abs/physics/9705023},
	doi = {10.1109/72.712178},
	abstract = {We present a method to solve initial and boundary value problems using artificial neural networks. A trial solution of the differential equation is written as a sum of two parts. The first part satisfies the boundary (or initial) conditions and contains no adjustable parameters. The second part is constructed so as not to affect the boundary conditions. This part involves a feedforward neural network, containing adjustable parameters (the weights). Hence by construction the boundary conditions are satisfied and the network is trained to satisfy the differential equation. The applicability of this approach ranges from single ODE's, to systems of coupled ODE's and also to PDE's. In this article we illustrate the method by solving a variety of model problems and present comparisons with finite elements for several cases of partial differential equations.},
	number = {5},
	urldate = {2022-10-18},
	journal = {IEEE Transactions on Neural Networks},
	author = {Lagaris, I. E. and Likas, A. and Fotiadis, D. I.},
	month = may,
	year = {1997},
	note = {arXiv: physics/9705023v1},
	keywords = {Collocation method, Finite elements, Neural networks, Neuroprocessors, Ordinary differential equations, Partial differential equations, notion, unread},
	pages = {987--1000},
}

@article{homan_no-u-turn_nodate,
	title = {The {No}-{U}-{Turn} {Sampler}: {Adaptively} {Setting} {Path} {Lengths} in {Hamiltonian} {Monte} {Carlo}},
	abstract = {Hamiltonian Monte Carlo (HMC) is a Markov chain Monte Carlo (MCMC) algorithm that avoids the random walk behavior and sensitivity to correlated parameters that plague many MCMC methods by taking a series of steps informed by ﬁrst-order gradient information. These features allow it to converge to high-dimensional target distributions much more quickly than simpler methods such as random walk Metropolis or Gibbs sampling. However, HMC’s performance is highly sensitive to two user-speciﬁed parameters: a step size and a desired number of steps L. In particular, if L is too small then the algorithm exhibits undesirable random walk behavior, while if L is too large the algorithm wastes computation. We introduce the No-U-Turn Sampler (NUTS), an extension to HMC that eliminates the need to set a number of steps L. NUTS uses a recursive algorithm to build a set of likely candidate points that spans a wide swath of the target distribution, stopping automatically when it starts to double back and retrace its steps. Empirically, NUTS performs at least as eﬃciently as (and sometimes more eﬃciently than) a well tuned standard HMC method, without requiring user intervention or costly tuning runs. We also derive a method for adapting the step size parameter on the ﬂy based on primal-dual averaging. NUTS can thus be used with no hand-tuning at all, making it suitable for applications such as BUGS-style automatic inference engines that require eﬃcient “turnkey” samplers.},
	language = {en},
	author = {Hoﬀman, Matthew D and Gelman, Andrew},
	keywords = {unread},
}

@article{hastings_monte_1970,
	title = {Monte {Carlo} {Sampling} {Methods} {Using} {Markov} {Chains} and {Their} {Applications}},
	volume = {57},
	issn = {0006-3444},
	url = {https://www.jstor.org/stable/2334940},
	doi = {10.2307/2334940},
	abstract = {A generalization of the sampling method introduced by Metropolis et al. (1953) is presented along with an exposition of the relevant theory, techniques of application and methods and difficulties of assessing the error in Monte Carlo estimates. Examples of the methods, including the generation of random orthogonal matrices and potential applications of the methods to numerical problems arising in statistics, are discussed.},
	number = {1},
	urldate = {2023-09-29},
	journal = {Biometrika},
	author = {Hastings, W. K.},
	year = {1970},
	note = {Publisher: [Oxford University Press, Biometrika Trust]},
	keywords = {unread},
	pages = {97--109},
}

@article{goodman_ensemble_2010,
	title = {Ensemble samplers with affine invariance},
	volume = {5},
	issn = {2157-5452},
	url = {https://msp.org/camcos/2010/5-1/p04.xhtml},
	doi = {10.2140/camcos.2010.5.65},
	number = {1},
	urldate = {2023-10-03},
	journal = {Communications in Applied Mathematics and Computational Science},
	author = {Goodman, Jonathan and Weare, Jonathan},
	month = jan,
	year = {2010},
	note = {Publisher: Mathematical Sciences Publishers},
	keywords = {unread},
	pages = {65--80},
}

@article{chen_neurodiffeq_2020,
	title = {{NeuroDiffEq}: {A} {Python} package for solving differential equations with neural networks},
	volume = {5},
	issn = {2475-9066},
	shorttitle = {{NeuroDiffEq}},
	url = {https://joss.theoj.org/papers/10.21105/joss.01931},
	doi = {10.21105/joss.01931},
	abstract = {Differential equations emerge in various scientific and engineering domains for modeling physical phenomena. Most differential equations of practical interest are analytically intractable. Traditionally, differential equations are solved by numerical methods. Sophisticated algorithms exist to integrate differential equations in time and space. Time integration techniques continue to be an active area of research and include backward difference formulas and Runge-Kutta methods (Conde, Gottlieb, Grant, \& Shadid, 2017). Common spatial discretization approaches include the finite difference method (FDM), finite volume method (FVM), and finite element method (FEM) as well as spectral methods such as the Fourier-spectral method. These classical methods have been studied in detail and much is known about their convergence properties. Moreover, highly optimized codes exist for solving differential equations of practical interest with these techniques (Seefeldt et al., 2017; Smith \& Abeysinghe, 2017). While these methods are efficient and well-studied, their expressibility is limited by their function representation.},
	language = {en},
	number = {46},
	urldate = {2023-09-28},
	journal = {Journal of Open Source Software},
	author = {Chen, Feiyu and Sondak, David and Protopapas, Pavlos and Mattheakis, Marios and Liu, Shuheng and Agarwal, Devansh and Di Giovanni, Marco},
	month = feb,
	year = {2020},
	keywords = {unread},
	pages = {1931},
}

@book{bishop_pattern_2006,
	address = {Berlin, Heidelberg},
	title = {Pattern {Recognition} and {Machine} {Learning} ({Information} {Science} and {Statistics})},
	isbn = {978-0-387-31073-2},
	publisher = {Springer-Verlag},
	author = {Bishop, Christopher M.},
	month = jul,
	year = {2006},
	keywords = {unread},
}

@misc{graf_error-aware_2022,
	title = {Error-{Aware} {B}-{PINNs}: {Improving} {Uncertainty} {Quantification} in {Bayesian} {Physics}-{Informed} {Neural} {Networks}},
	shorttitle = {Error-{Aware} {B}-{PINNs}},
	url = {http://arxiv.org/abs/2212.06965},
	doi = {10.48550/arXiv.2212.06965},
	abstract = {Physics-Informed Neural Networks (PINNs) are gaining popularity as a method for solving differential equations. While being more feasible in some contexts than the classical numerical techniques, PINNs still lack credibility. A remedy for that can be found in Uncertainty Quantification (UQ) which is just beginning to emerge in the context of PINNs. Assessing how well the trained PINN complies with imposed differential equation is the key to tackling uncertainty, yet there is lack of comprehensive methodology for this task. We propose a framework for UQ in Bayesian PINNs (B-PINNs) that incorporates the discrepancy between the B-PINN solution and the unknown true solution. We exploit recent results on error bounds for PINNs on linear dynamical systems and demonstrate the predictive uncertainty on a class of linear ODEs.},
	urldate = {2023-03-17},
	publisher = {arXiv},
	author = {Graf, Olga and Flores, Pablo and Protopapas, Pavlos and Pichara, Karim},
	month = dec,
	year = {2022},
	note = {arXiv:2212.06965 [cs]},
	keywords = {Computer Science - Machine Learning, unread},
}

@misc{graf_uncertainty_2021,
	title = {Uncertainty {Quantification} in {Neural} {Differential} {Equations}},
	url = {http://arxiv.org/abs/2111.04207},
	doi = {10.48550/arXiv.2111.04207},
	abstract = {Uncertainty quantification (UQ) helps to make trustworthy predictions based on collected observations and uncertain domain knowledge. With increased usage of deep learning in various applications, the need for efficient UQ methods that can make deep models more reliable has increased as well. Among applications that can benefit from effective handling of uncertainty are the deep learning based differential equation (DE) solvers. We adapt several state-of-the-art UQ methods to get the predictive uncertainty for DE solutions and show the results on four different DE types.},
	urldate = {2022-12-14},
	publisher = {arXiv},
	author = {Graf, Olga and Flores, Pablo and Protopapas, Pavlos and Pichara, Karim},
	month = nov,
	year = {2021},
	note = {arXiv:2111.04207 [cs]},
	keywords = {Computer Science - Machine Learning, unread},
}

@article{foreman-mackey_emcee_2013,
	title = {emcee: {The} {MCMC} {Hammer}},
	volume = {125},
	issn = {1538-3873},
	shorttitle = {emcee},
	url = {https://iopscience.iop.org/article/10.1086/670067/meta},
	doi = {10.1086/670067},
	language = {en},
	number = {925},
	urldate = {2023-10-03},
	journal = {Publications of the Astronomical Society of the Pacific},
	author = {Foreman-Mackey, Daniel and Hogg, David W. and Lang, Dustin and Goodman, Jonathan},
	month = feb,
	year = {2013},
	note = {Publisher: IOP Publishing},
	keywords = {unread},
	pages = {306},
}

@misc{flamant_solving_2020,
	title = {Solving {Differential} {Equations} {Using} {Neural} {Network} {Solution} {Bundles}},
	url = {https://arxiv.org/abs/2006.14372v1},
	abstract = {The time evolution of dynamical systems is frequently described by ordinary differential equations (ODEs), which must be solved for given initial conditions. Most standard approaches numerically integrate ODEs producing a single solution whose values are computed at discrete times. When many varied solutions with different initial conditions to the ODE are required, the computational cost can become significant. We propose that a neural network be used as a solution bundle, a collection of solutions to an ODE for various initial states and system parameters. The neural network solution bundle is trained with an unsupervised loss that does not require any prior knowledge of the sought solutions, and the resulting object is differentiable in initial conditions and system parameters. The solution bundle exhibits fast, parallelizable evaluation of the system state, facilitating the use of Bayesian inference for parameter estimation in real dynamical systems.},
	language = {en},
	urldate = {2023-09-28},
	journal = {arXiv.org},
	author = {Flamant, Cedric and Protopapas, Pavlos and Sondak, David},
	month = jun,
	year = {2020},
	keywords = {unread},
}

@article{duane_hybrid_1987,
	title = {Hybrid {Monte} {Carlo}},
	volume = {195},
	issn = {0370-2693},
	url = {https://www.sciencedirect.com/science/article/pii/037026938791197X},
	doi = {10.1016/0370-2693(87)91197-X},
	abstract = {We present a new method for the numerical simulation of lattice field theory. A hybrid (molecular dynamics/Langevin) algorithm is used to guide a Monte Carlo simulation. There are no discretization errors even for large step sizes. The method is especially efficient for systems such as quantum chromodynamics which contain fermionic degrees of freedom. Detailed results are presented for four-dimensional compact quantum electrodynamics including the dynamical effects of electrons.},
	number = {2},
	urldate = {2023-09-28},
	journal = {Physics Letters B},
	author = {Duane, Simon and Kennedy, A. D. and Pendleton, Brian J. and Roweth, Duncan},
	month = sep,
	year = {1987},
	keywords = {unread},
	pages = {216--222},
}

@article{blundell_weight_2015,
	title = {Weight uncertainty in neural networks},
	volume = {2},
	abstract = {We introduce a new, efficient, principled and backpropagation-compatible algorithm for learning a probability distribution on the weights of a neural network, called Bayes by Backprop. It regularises the weights by minimising a compression cost, known as the variational free energy or the expected lower bound on the marginal likelihood. We show that this principled kind of regularisation yields comparable performance to dropout on MNIST classification. We then demonstrate how the learnt uncertainty in the weights can be used to improve generalisation in non-linear regression problems, and how this weight uncertainty can be used to drive the exploration-exploitation trade-off in reinforcement learning.},
	journal = {32nd International Conference on Machine Learning, ICML 2015},
	author = {Blundell, Charles and Cornebise, Julien and Kavukcuoglu, Koray and Wierstra, Daan},
	year = {2015},
	note = {arXiv: 1505.05424
ISBN: 9781510810587},
	keywords = {notion, unread},
	pages = {1613--1622},
}

@misc{graf_uncertainty_2021-1,
	title = {Uncertainty {Quantification} in {Neural} {Differential} {Equations}},
	url = {http://arxiv.org/abs/2111.04207},
	doi = {10.48550/arXiv.2111.04207},
	abstract = {Uncertainty quantification (UQ) helps to make trustworthy predictions based on collected observations and uncertain domain knowledge. With increased usage of deep learning in various applications, the need for efficient UQ methods that can make deep models more reliable has increased as well. Among applications that can benefit from effective handling of uncertainty are the deep learning based differential equation (DE) solvers. We adapt several state-of-the-art UQ methods to get the predictive uncertainty for DE solutions and show the results on four different DE types.},
	urldate = {2023-10-06},
	publisher = {arXiv},
	author = {Graf, Olga and Flores, Pablo and Protopapas, Pavlos and Pichara, Karim},
	month = nov,
	year = {2021},
	note = {arXiv:2111.04207 [cs]},
	keywords = {Computer Science - Machine Learning, unread},
}

@article{cuomo_scientific_2022,
	title = {Scientific {Machine} {Learning} {Through} {Physics}–{Informed} {Neural} {Networks}: {Where} we are and {What}’s {Next}},
	volume = {92},
	issn = {1573-7691},
	shorttitle = {Scientific {Machine} {Learning} {Through} {Physics}–{Informed} {Neural} {Networks}},
	url = {https://doi.org/10.1007/s10915-022-01939-z},
	doi = {10.1007/s10915-022-01939-z},
	abstract = {Physics-Informed Neural Networks (PINN) are neural networks (NNs) that encode model equations, like Partial Differential Equations (PDE), as a component of the neural network itself. PINNs are nowadays used to solve PDEs, fractional equations, integral-differential equations, and stochastic PDEs. This novel methodology has arisen as a multi-task learning framework in which a NN must fit observed data while reducing a PDE residual. This article provides a comprehensive review of the literature on PINNs: while the primary goal of the study was to characterize these networks and their related advantages and disadvantages. The review also attempts to incorporate publications on a broader range of collocation-based physics informed neural networks, which stars form the vanilla PINN, as well as many other variants, such as physics-constrained neural networks (PCNN), variational hp-VPINN, and conservative PINN (CPINN). The study indicates that most research has focused on customizing the PINN through different activation functions, gradient optimization techniques, neural network structures, and loss function structures. Despite the wide range of applications for which PINNs have been used, by demonstrating their ability to be more feasible in some contexts than classical numerical techniques like Finite Element Method (FEM), advancements are still possible, most notably theoretical issues that remain unresolved.},
	language = {en},
	number = {3},
	urldate = {2023-10-06},
	journal = {Journal of Scientific Computing},
	author = {Cuomo, Salvatore and Di Cola, Vincenzo Schiano and Giampaolo, Fabio and Rozza, Gianluigi and Raissi, Maziar and Piccialli, Francesco},
	month = jul,
	year = {2022},
	keywords = {Deep Neural Networks, Nonlinear equations, Numerical methods, Partial Differential Equations, Physics–Informed Neural Networks, Scientific Machine Learning, Uncertainty, unread},
	pages = {88},
}

@article{raissi_physics-informed_2019-1,
	title = {Physics-informed neural networks: {A} deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations},
	volume = {378},
	issn = {10902716},
	doi = {10.1016/j.jcp.2018.10.045},
	abstract = {We introduce physics-informed neural networks – neural networks that are trained to solve supervised learning tasks while respecting any given laws of physics described by general nonlinear partial differential equations. In this work, we present our developments in the context of solving two main classes of problems: data-driven solution and data-driven discovery of partial differential equations. Depending on the nature and arrangement of the available data, we devise two distinct types of algorithms, namely continuous time and discrete time models. The first type of models forms a new family of data-efficient spatio-temporal function approximators, while the latter type allows the use of arbitrarily accurate implicit Runge–Kutta time stepping schemes with unlimited number of stages. The effectiveness of the proposed framework is demonstrated through a collection of classical problems in fluids, quantum mechanics, reaction–diffusion systems, and the propagation of nonlinear shallow-water waves.},
	journal = {Journal of Computational Physics},
	author = {Raissi, M. and Perdikaris, P. and Karniadakis, G. E.},
	month = feb,
	year = {2019},
	note = {Publisher: Academic Press Inc.},
	keywords = {Data-driven scientific computing, Machine learning, Nonlinear dynamics, Predictive modeling, Runge–Kutta methods, notion, unread},
	pages = {686--707},
}

@misc{zhang_physics-informed_2020,
	title = {Physics-{Informed} {Neural} {Networks} for {Nonhomogeneous} {Material} {Identification} in {Elasticity} {Imaging}},
	url = {http://arxiv.org/abs/2009.04525},
	doi = {10.48550/arXiv.2009.04525},
	abstract = {We apply Physics-Informed Neural Networks (PINNs) for solving identification problems of nonhomogeneous materials. We focus on the problem with a background in elasticity imaging, where one seeks to identify the nonhomogeneous mechanical properties of soft tissue based on the full-field displacement measurements under quasi-static loading. In our model, we apply two independent neural networks, one for approximating the solution of the corresponding forward problem, and the other for approximating the unknown material parameter field. As a proof of concept, we validate our model on a prototypical plane strain problem for incompressible hyperelastic tissue. The results show that the PINNs are effective in accurately recovering the unknown distribution of mechanical properties. By employing two neural networks in our model, we extend the capability of material identification of PINNs to include nonhomogeneous material parameter fields, which enables more flexibility of PINNs in representing complex material properties.},
	urldate = {2022-11-09},
	publisher = {arXiv},
	author = {Zhang, Enrui and Yin, Minglang and Karniadakis, George Em},
	month = sep,
	year = {2020},
	note = {arXiv:2009.04525 [cond-mat]},
	keywords = {Computer Science - Machine Learning, Condensed Matter - Soft Condensed Matter, Electrical Engineering and Systems Science - Image and Video Processing, unread},
}

@article{mao_physics-informed_2020,
	title = {Physics-informed neural networks for high-speed flows},
	volume = {360},
	issn = {0045-7825},
	url = {https://www.sciencedirect.com/science/article/pii/S0045782519306814},
	doi = {10.1016/j.cma.2019.112789},
	abstract = {In this work we investigate the possibility of using physics-informed neural networks (PINNs) to approximate the Euler equations that model high-speed aerodynamic flows. In particular, we solve both the forward and inverse problems in one-dimensional and two-dimensional domains. For the forward problem, we utilize the Euler equations and the initial/boundary conditions to formulate the loss function, and solve the one-dimensional Euler equations with smooth solutions and with solutions that have a contact discontinuity as well as a two-dimensional oblique shock wave problem. We demonstrate that we can capture the solutions with only a few scattered points clustered randomly around the discontinuities. For the inverse problem, motivated by mimicking the Schlieren photography experimental technique used traditionally in high-speed aerodynamics, we use the data on density gradient ∇ρ(x,t), the pressure p(x∗,t) at a specified point x=x∗ as well as the conservation laws to infer all states of interest (density, velocity and pressure fields). We present illustrative benchmark examples for both the problem with smooth solutions and Riemann problems (Sod and Lax problems) with PINNs, demonstrating that all inferred states are in good agreement with the reference solutions. Moreover, we show that the choice of the position of the point x∗ plays an important role in the learning process. In particular, for the problem with smooth solutions we can randomly choose the position of the point x∗ from the computational domain, while for the Sod or Lax problem, we have to choose the position of the point x∗ from the domain between the initial discontinuous point and the shock position of the final time. We also solve the inverse problem by combining the aforementioned data and the Euler equations in characteristic form, showing that the results obtained by using the Euler equations in characteristic form are better than that obtained by using the Euler equations in conservative form. Furthermore, we consider another type of inverse problem, specifically, we employ PINNs to learn the value of the parameter γ in the equation of state for the parameterized two-dimensional oblique wave problem by using the given data of the density, velocity and the pressure, and we identify the parameter γ accurately. Taken together, our results demonstrate that in the current form, where the conservation laws are imposed at random points, PINNs are not as accurate as traditional numerical methods for forward problems but they are superior for inverse problems that cannot even be solved with standard techniques.},
	language = {en},
	urldate = {2022-11-21},
	journal = {Computer Methods in Applied Mechanics and Engineering},
	author = {Mao, Zhiping and Jagtap, Ameya D. and Karniadakis, George Em},
	month = mar,
	year = {2020},
	keywords = {Conservation laws, Euler equations, Hidden fluid mechanics, Machine learning, Neural networks, Riemann problem, unread},
	pages = {112789},
}

@article{cai_physics-informed_2021,
	title = {Physics-{Informed} {Neural} {Networks} for {Heat} {Transfer} {Problems}},
	volume = {143},
	issn = {0022-1481},
	url = {https://doi.org/10.1115/1.4050542},
	doi = {10.1115/1.4050542},
	abstract = {Physics-informed neural networks (PINNs) have gained popularity across different engineering fields due to their effectiveness in solving realistic problems with noisy data and often partially missing physics. In PINNs, automatic differentiation is leveraged to evaluate differential operators without discretization errors, and a multitask learning problem is defined in order to simultaneously fit observed data while respecting the underlying governing laws of physics. Here, we present applications of PINNs to various prototype heat transfer problems, targeting in particular realistic conditions not readily tackled with traditional computational methods. To this end, we first consider forced and mixed convection with unknown thermal boundary conditions on the heated surfaces and aim to obtain the temperature and velocity fields everywhere in the domain, including the boundaries, given some sparse temperature measurements. We also consider the prototype Stefan problem for two-phase flow, aiming to infer the moving interface, the velocity and temperature fields everywhere as well as the different conductivities of a solid and a liquid phase, given a few temperature measurements inside the domain. Finally, we present some realistic industrial applications related to power electronics to highlight the practicality of PINNs as well as the effective use of neural networks in solving general heat transfer problems of industrial complexity. Taken together, the results presented herein demonstrate that PINNs not only can solve ill-posed problems, which are beyond the reach of traditional computational methods, but they can also bridge the gap between computational and experimental heat transfer.},
	number = {6},
	urldate = {2022-11-21},
	journal = {Journal of Heat Transfer},
	author = {Cai, Shengze and Wang, Zhicheng and Wang, Sifan and Perdikaris, Paris and Karniadakis, George Em},
	month = apr,
	year = {2021},
	keywords = {unread},
}

@article{sahli_costabal_physics-informed_2020,
	title = {Physics-{Informed} {Neural} {Networks} for {Cardiac} {Activation} {Mapping}},
	volume = {8},
	issn = {2296-424X},
	url = {https://www.frontiersin.org/articles/10.3389/fphy.2020.00042},
	abstract = {A critical procedure in diagnosing atrial fibrillation is the creation of electro-anatomic activation maps. Current methods generate these mappings from interpolation using a few sparse data points recorded inside the atria; they neither include prior knowledge of the underlying physics nor uncertainty of these recordings. Here we propose a physics-informed neural network for cardiac activation mapping that accounts for the underlying wave propagation dynamics and we quantify the epistemic uncertainty associated with these predictions. These uncertainty estimates not only allow us to quantify the predictive error of the neural network, but also help to reduce it by judiciously selecting new informative measurement locations via active learning. We illustrate the potential of our approach using a synthetic benchmark problem and a personalized electrophysiology model of the left atrium. We show that our new method outperforms linear interpolation and Gaussian process regression for the benchmark problem and linear interpolation at clinical densities for the left atrium. In both cases, the active learning algorithm achieves lower error levels than random allocation. Our findings open the door toward physics-based electro-anatomic mapping with the ultimate goals to reduce procedural time and improve diagnostic predictability for patients affected by atrial fibrillation. Open source code is available at https://github.com/fsahli/EikonalNet.},
	urldate = {2022-11-21},
	journal = {Frontiers in Physics},
	author = {Sahli Costabal, Francisco and Yang, Yibo and Perdikaris, Paris and Hurtado, Daniel E. and Kuhl, Ellen},
	year = {2020},
	keywords = {notion, unread},
}

@article{cai_physics-informed_2021-1,
	title = {Physics-informed neural networks ({PINNs}) for fluid mechanics: a review},
	volume = {37},
	issn = {1614-3116},
	shorttitle = {Physics-informed neural networks ({PINNs}) for fluid mechanics},
	url = {https://doi.org/10.1007/s10409-021-01148-1},
	doi = {10.1007/s10409-021-01148-1},
	abstract = {Despite the significant progress over the last 50 years in simulating flow problems using numerical discretization of the Navier–Stokes equations (NSE), we still cannot incorporate seamlessly noisy data into existing algorithms, mesh-generation is complex, and we cannot tackle high-dimensional problems governed by parametrized NSE. Moreover, solving inverse flow problems is often prohibitively expensive and requires complex and expensive formulations and new computer codes. Here, we review flow physics-informed learning, integrating seamlessly data and mathematical models, and implement them using physics-informed neural networks (PINNs). We demonstrate the effectiveness of PINNs for inverse problems related to three-dimensional wake flows, supersonic flows, and biomedical flows.},
	language = {en},
	number = {12},
	urldate = {2022-11-22},
	journal = {Acta Mechanica Sinica},
	author = {Cai, Shengze and Mao, Zhiping and Wang, Zhicheng and Yin, Minglang and Karniadakis, George Em},
	month = dec,
	year = {2021},
	keywords = {Biomedical flows, Inverse problems, PINNs, Physics-informed learning, Supersonic flows, notion, unread},
	pages = {1727--1738},
}

@article{rasht-behesht_physics-informed_2022,
	title = {Physics-{Informed} {Neural} {Networks} ({PINNs}) for {Wave} {Propagation} and {Full} {Waveform} {Inversions}},
	volume = {127},
	issn = {2169-9356},
	url = {https://onlinelibrary.wiley.com/doi/abs/10.1029/2021JB023120},
	doi = {10.1029/2021JB023120},
	abstract = {We propose a new approach to the solution of the wave propagation and full waveform inversions (FWIs) based on a recent advance in deep learning called physics-informed neural networks (PINNs). In this study, we present an algorithm for PINNs applied to the acoustic wave equation and test the method with both forward models and FWI case studies. These synthetic case studies are designed to explore the ability of PINNs to handle varying degrees of structural complexity using both teleseismic plane waves and seismic point sources. PINNs' meshless formalism allows for a flexible implementation of the wave equation and different types of boundary conditions. For instance, our models demonstrate that PINN automatically satisfies absorbing boundary conditions, a serious computational challenge for common wave propagation solvers. Furthermore, a priori knowledge of the subsurface structure can be seamlessly encoded in PINNs' formulation. We find that the current state-of-the-art PINNs provide good results for the forward model, even though spectral element or finite difference methods are more efficient and accurate. More importantly, our results demonstrate that PINNs yield excellent results for inversions on all cases considered and with limited computational complexity. We discuss the current limitations of the method with complex velocity models as well as strategies to overcome these challenges. Using PINNs as a geophysical inversion solver offers exciting perspectives, not only for the full waveform seismic inversions, but also when dealing with other geophysical datasets (e.g., MT, gravity) as well as joint inversions because of its robust framework and simple implementation.},
	language = {en},
	number = {5},
	urldate = {2022-11-09},
	journal = {Journal of Geophysical Research: Solid Earth},
	author = {Rasht-Behesht, Majid and Huber, Christian and Shukla, Khemraj and Karniadakis, George Em},
	year = {2022},
	note = {\_eprint: https://onlinelibrary.wiley.com/doi/pdf/10.1029/2021JB023120},
	keywords = {acoustic wave propagation, deep learning, full waveform inversion, physics-informed neural networks, unread},
	pages = {e2021JB023120},
}

@article{jin_nsfnets_2021,
	title = {{NSFnets} ({Navier}-{Stokes} flow nets): {Physics}-informed neural networks for the incompressible {Navier}-{Stokes} equations},
	volume = {426},
	issn = {0021-9991},
	shorttitle = {{NSFnets} ({Navier}-{Stokes} flow nets)},
	url = {https://www.sciencedirect.com/science/article/pii/S0021999120307257},
	doi = {10.1016/j.jcp.2020.109951},
	abstract = {In the last 50 years there has been a tremendous progress in solving numerically the Navier-Stokes equations using finite differences, finite elements, spectral, and even meshless methods. Yet, in many real cases, we still cannot incorporate seamlessly (multi-fidelity) data into existing algorithms, and for industrial-complexity applications the mesh generation is time consuming and still an art. Moreover, solving ill-posed problems (e.g., lacking boundary conditions) or inverse problems is often prohibitively expensive and requires different formulations and new computer codes. Here, we employ physics-informed neural networks (PINNs), encoding the governing equations directly into the deep neural network via automatic differentiation, to overcome some of the aforementioned limitations for simulating incompressible laminar and turbulent flows. We develop the Navier-Stokes flow nets (NSFnets) by considering two different mathematical formulations of the Navier-Stokes equations: the velocity-pressure (VP) formulation and the vorticity-velocity (VV) formulation. Since this is a new approach, we first select some standard benchmark problems to assess the accuracy, convergence rate, computational cost and flexibility of NSFnets; analytical solutions and direct numerical simulation (DNS) databases provide proper initial and boundary conditions for the NSFnet simulations. The spatial and temporal coordinates are the inputs of the NSFnets, while the instantaneous velocity and pressure fields are the outputs for the VP-NSFnet, and the instantaneous velocity and vorticity fields are the outputs for the VV-NSFnet. This is unsupervised learning and, hence, no labeled data are required beyond boundary and initial conditions and the fluid properties. The residuals of the VP or VV governing equations, together with the initial and boundary conditions, are embedded into the loss function of the NSFnets. No data is provided for the pressure to the VP-NSFnet, which is a hidden state and is obtained via the incompressibility constraint without extra computational cost. Unlike the traditional numerical methods, NSFnets inherit the properties of neural networks (NNs), hence the total error is composed of the approximation, the optimization, and the generalization errors. Here, we empirically attempt to quantify these errors by varying the sampling (“residual”) points, the iterative solvers, and the size of the NN architecture. For the laminar flow solutions, we show that both the VP and the VV formulations are comparable in accuracy but their best performance corresponds to different NN architectures. The initial convergence rate is fast but the error eventually saturates to a plateau due to the dominance of the optimization error. For the turbulent channel flow, we show that NSFnets can sustain turbulence at Reτ∼1,000, but due to expensive training we only consider part of the channel domain and enforce velocity boundary conditions on the subdomain boundaries provided by the DNS data base. We also perform a systematic study on the weights used in the loss function for balancing the data and physics components, and investigate a new way of computing the weights dynamically to accelerate training and enhance accuracy. In the last part, we demonstrate how NSFnets should be used in practice, namely for ill-posed problems with incomplete or noisy boundary conditions as well as for inverse problems. We obtain reasonably accurate solutions for such cases as well without the need to change the NSFnets and at the same computational cost as in the forward well-posed problems. We also present a simple example of transfer learning that will aid in accelerating the training of NSFnets for different parameter settings.},
	language = {en},
	urldate = {2022-11-21},
	journal = {Journal of Computational Physics},
	author = {Jin, Xiaowei and Cai, Shengze and Li, Hui and Karniadakis, George Em},
	month = feb,
	year = {2021},
	keywords = {Ill-posed problems, PINNs, Transfer learning, Turbulence, Velocity-pressure formulation, Vorticity-velocity formulation, unread},
	pages = {109951},
}

@misc{daw_mitigating_2022,
	title = {Mitigating {Propagation} {Failures} in {PINNs} using {Evolutionary} {Sampling}},
	url = {http://arxiv.org/abs/2207.02338},
	doi = {10.48550/arXiv.2207.02338},
	abstract = {Despite the success of physics-informed neural networks (PINNs) in approximating partial differential equations (PDEs), it is known that PINNs can sometimes fail to converge to the correct solution in problems involving complicated PDEs. This is reflected in several recent studies on characterizing and mitigating the ``failure modes'' of PINNs. While most of these studies have focused on balancing loss functions or adaptively tuning PDE coefficients, what is missing is a thorough understanding of the connection between failure modes of PINNs and sampling strategies used for training PINNs. In this paper, we provide a novel perspective of failure modes of PINNs by hypothesizing that the training of PINNs rely on successful ``propagation'' of solution from initial and/or boundary condition points to interior points. We show that PINNs with poor sampling strategies can get stuck at trivial solutions if there are propagation failures. We additionally demonstrate that propagation failures are characterized by highly imbalanced PDE residual fields where very high residuals are observed over very narrow regions. To mitigate propagation failures, we propose a novel evolutionary sampling (Evo) method that can incrementally accumulate collocation points in regions of high PDE residuals with little to no computational overhead. We provide an extension of Evo to respect the principle of causality while solving time-dependent PDEs. We theoretically analyze the behavior of Evo and empirically demonstrate its efficacy and efficiency in comparison with baselines on a variety of PDE problems.},
	urldate = {2022-11-21},
	publisher = {arXiv},
	author = {Daw, Arka and Bu, Jie and Wang, Sifan and Perdikaris, Paris and Karpatne, Anuj},
	month = oct,
	year = {2022},
	note = {arXiv:2207.02338 [cs]},
	keywords = {Computer Science - Artificial Intelligence, Computer Science - Machine Learning, To Read, notion, unread},
}

@article{de_ryck_error_2022,
	title = {Error analysis for physics-informed neural networks ({PINNs}) approximating {Kolmogorov} {PDEs}},
	volume = {48},
	issn = {1572-9044},
	url = {https://doi.org/10.1007/s10444-022-09985-9},
	doi = {10.1007/s10444-022-09985-9},
	abstract = {Physics-informed neural networks approximate solutions of PDEs by minimizing pointwise residuals. We derive rigorous bounds on the error, incurred by PINNs in approximating the solutions of a large class of linear parabolic PDEs, namely Kolmogorov equations that include the heat equation and Black-Scholes equation of option pricing, as examples. We construct neural networks, whose PINN residual (generalization error) can be made as small as desired. We also prove that the total L2-error can be bounded by the generalization error, which in turn is bounded in terms of the training error, provided that a sufficient number of randomly chosen training (collocation) points is used. Moreover, we prove that the size of the PINNs and the number of training samples only grow polynomially with the underlying dimension, enabling PINNs to overcome the curse of dimensionality in this context. These results enable us to provide a comprehensive error analysis for PINNs in approximating Kolmogorov PDEs.},
	language = {en},
	number = {6},
	urldate = {2022-11-21},
	journal = {Advances in Computational Mathematics},
	author = {De Ryck, Tim and Mishra, Siddhartha},
	month = nov,
	year = {2022},
	keywords = {65M99, Deep learning, Kolmogorov equations, Physics-informed neural networks, unread},
	pages = {79},
}

@misc{misyris_capturing_2021,
	title = {Capturing {Power} {System} {Dynamics} by {Physics}-{Informed} {Neural} {Networks} and {Optimization}},
	url = {http://arxiv.org/abs/2103.17004},
	doi = {10.48550/arXiv.2103.17004},
	abstract = {This paper proposes a tractable framework to determine key characteristics of non-linear dynamic systems by converting physics-informed neural networks to a mixed integer linear program. Our focus is on power system applications. Traditional methods in power systems require the use of a large number of simulations and other heuristics to determine parameters such as the critical clearing time, i.e. the maximum allowable time within which a disturbance must be cleared before the system moves to instability. The work proposed in this paper uses physics-informed neural networks to capture the power system dynamic behavior and, through an exact transformation, converts them to a tractable optimization problem which can be used to determine critical system indices. By converting neural networks to mixed integer linear programs, our framework also allows to adjust the conservativeness of the neural network output with respect to the existing stability boundaries. We demonstrate the performance of our methods on the non-linear dynamics of converter-based generation in response to voltage disturbances.},
	urldate = {2022-11-23},
	publisher = {arXiv},
	author = {Misyris, Georgios S. and Stiasny, Jochen and Chatzivasileiadis, Spyros},
	month = mar,
	year = {2021},
	note = {arXiv:2103.17004 [cs, eess]},
	keywords = {Electrical Engineering and Systems Science - Systems and Control, unread},
}

@inproceedings{steger_how_2022,
	title = {How {PINNs} cheat: {Predicting} chaotic motion of a double pendulum},
	shorttitle = {How {PINNs} cheat},
	url = {https://openreview.net/forum?id=shUbBca03f},
	abstract = {Despite extensive research, physics-informed neural networks (PINNs) are still difficult to train, especially when the optimization relies heavily on the physics loss term. Convergence problems frequently occur when simulating dynamical systems with high-frequency components, chaotic or turbulent behavior. In this work, we discuss whether the traditional PINN framework is able to predict chaotic motion by conducting experiments on the undamped double pendulum. Our results demonstrate that PINNs do not exhibit any sensitivity to perturbations in the initial condition. Instead, the PINN optimization consistently converges to physically correct solutions that violate the initial condition only marginally, but diverge significantly from the desired solution due to the chaotic nature of the system. In fact, the PINN predictions primarily exhibit low-frequency components with a smaller magnitude of higher-order derivatives, which favors lower physics loss values compared to the desired solution. We thus hypothesize that the PINNs "cheat" by shifting the initial conditions to values that correspond to physically correct solutions that are easier to learn. Initial experiments suggest that domain decomposition combined with an appropriate loss weighting scheme mitigates this effect and allows convergence to the desired solution.},
	language = {en},
	urldate = {2022-11-07},
	author = {Steger, Sophie and Rohrhofer, Franz M. and Geiger, Bernhard C.},
	month = oct,
	year = {2022},
	keywords = {unread},
}

@inproceedings{de_ryck_generic_2022,
	title = {Generic bounds on the approximation error for physics-informed (and) operator learning},
	url = {https://openreview.net/forum?id=bF4eYy3LTR9},
	abstract = {We propose a very general framework for deriving rigorous bounds on the approximation error for physics-informed neural networks (PINNs) and operator learning architectures such as DeepONets and FNOs as well as for physics-informed operator learning. These bounds guarantee that PINNs and (physics-informed) DeepONets or FNOs will efficiently approximate the underlying solution or solution-operator of generic partial differential equations (PDEs). Our framework utilizes existing neural network approximation results to obtain bounds on more-involved learning architectures for PDEs. We illustrate the general framework by deriving the first rigorous bounds on the approximation error of physics-informed operator learning and by showing that PINNs (and physics-informed DeepONets and FNOs) mitigate the curse of dimensionality in approximating nonlinear parabolic PDEs.},
	language = {en},
	urldate = {2023-10-08},
	author = {De Ryck, Tim and Mishra, Siddhartha},
	month = may,
	year = {2022},
	keywords = {unread},
}

@misc{de_ryck_error_2022-1,
	title = {Error estimates for physics informed neural networks approximating the {Navier}-{Stokes} equations},
	url = {http://arxiv.org/abs/2203.09346},
	doi = {10.48550/arXiv.2203.09346},
	abstract = {We prove rigorous bounds on the errors resulting from the approximation of the incompressible Navier-Stokes equations with (extended) physics informed neural networks. We show that the underlying PDE residual can be made arbitrarily small for tanh neural networks with two hidden layers. Moreover, the total error can be estimated in terms of the training error, network size and number of quadrature points. The theory is illustrated with numerical experiments.},
	urldate = {2022-11-21},
	publisher = {arXiv},
	author = {De Ryck, Tim and Jagtap, Ameya D. and Mishra, Siddhartha},
	month = mar,
	year = {2022},
	note = {arXiv:2203.09346 [cs, math]},
	keywords = {Computer Science - Machine Learning, Mathematics - Numerical Analysis, unread},
}

@article{guo_energy-based_2022,
	title = {Energy-{Based} {Error} {Bound} of {Physics}-{Informed} {Neural} {Network} {Solutions} in {Elasticity}},
	volume = {148},
	copyright = {© 2022 American Society of Civil Engineers},
	issn = {1943-7889},
	url = {https://ascelibrary.org/doi/10.1061/%28ASCE%29EM.1943-7889.0002121},
	doi = {10.1061/(ASCE)EM.1943-7889.0002121},
	abstract = {An energy-based a posteriori error bound is proposed for the physics-informed neural network solutions of elasticity problems. An admissible displacement-stress solution pair is obtained from a mixed form of physics-informed neural networks, and the proposed error bound is formulated as the constitutive relation error defined by the solution pair. Such an error estimator provides an upper bound of the global error of neural network discretization. The bounding property, as well as the asymptotic behavior of the physics-informed neural network solutions, are studied in a demonstration example.},
	language = {EN},
	number = {8},
	urldate = {2022-11-21},
	journal = {Journal of Engineering Mechanics},
	author = {Guo, Mengwu and Haghighat, Ehsan},
	month = aug,
	year = {2022},
	note = {Publisher: American Society of Civil Engineers},
	keywords = {unread},
	pages = {04022038},
}

@article{nabian_efficient_2021,
	title = {Efficient training of physics-informed neural networks via importance sampling},
	volume = {36},
	issn = {1467-8667},
	url = {https://onlinelibrary.wiley.com/doi/abs/10.1111/mice.12685},
	doi = {10.1111/mice.12685},
	abstract = {Physics-informed neural networks (PINNs) are a class of deep neural networks that are trained, using automatic differentiation, to compute the response of systems governed by partial differential equations (PDEs). The training of PINNs is simulation free, and does not require any training data set to be obtained from numerical PDE solvers. Instead, it only requires the physical problem description, including the governing laws of physics, domain geometry, initial/boundary conditions, and the material properties. This training usually involves solving a nonconvex optimization problem using variants of the stochastic gradient descent method, with the gradient of the loss function approximated on a batch of collocation points, selected randomly in each iteration according to a uniform distribution. Despite the success of PINNs in accurately solving a wide variety of PDEs, the method still requires improvements in terms of computational efficiency. To this end, in this paper, we study the performance of an importance sampling approach for efficient training of PINNs. Using numerical examples together with theoretical evidences, we show that in each training iteration, sampling the collocation points according to a distribution proportional to the loss function will improve the convergence behavior of the PINNs training. Additionally, we show that providing a piecewise constant approximation to the loss function for faster importance sampling can further improve the training efficiency. This importance sampling approach is straightforward and easy to implement in the existing PINN codes, and also does not introduce any new hyperparameter to calibrate. The numerical examples include elasticity, diffusion, and plane stress problems, through which we numerically verify the accuracy and efficiency of the importance sampling approach compared to the predominant uniform sampling approach.},
	language = {en},
	number = {8},
	urldate = {2022-11-21},
	journal = {Computer-Aided Civil and Infrastructure Engineering},
	author = {Nabian, Mohammad Amin and Gladstone, Rini Jasmine and Meidani, Hadi},
	year = {2021},
	note = {\_eprint: https://onlinelibrary.wiley.com/doi/pdf/10.1111/mice.12685},
	keywords = {unread},
	pages = {962--977},
}

@misc{linka_bayesian_2022,
	title = {Bayesian {Physics}-{Informed} {Neural} {Networks} for real-world nonlinear dynamical systems},
	url = {http://arxiv.org/abs/2205.08304},
	doi = {10.48550/arXiv.2205.08304},
	abstract = {Understanding real-world dynamical phenomena remains a challenging task. Across various scientific disciplines, machine learning has advanced as the go-to technology to analyze nonlinear dynamical systems, identify patterns in big data, and make decision around them. Neural networks are now consistently used as universal function approximators for data with underlying mechanisms that are incompletely understood or exceedingly complex. However, neural networks alone ignore the fundamental laws of physics and often fail to make plausible predictions. Here we integrate data, physics, and uncertainties by combining neural networks, physics-informed modeling, and Bayesian inference to improve the predictive potential of traditional neural network models. We embed the physical model of a damped harmonic oscillator into a fully-connected feed-forward neural network to explore a simple and illustrative model system, the outbreak dynamics of COVID-19. Our Physics-Informed Neural Networks can seamlessly integrate data and physics, robustly solve forward and inverse problems, and perform well for both interpolation and extrapolation, even for a small amount of noisy and incomplete data. At only minor additional cost, they can self-adaptively learn the weighting between data and physics. Combined with Bayesian Neural Networks, they can serve as priors in a Bayesian Inference, and provide credible intervals for uncertainty quantification. Our study reveals the inherent advantages and disadvantages of Neural Networks, Bayesian Inference, and a combination of both and provides valuable guidelines for model selection. While we have only demonstrated these approaches for the simple model problem of a seasonal endemic infectious disease, we anticipate that the underlying concepts and trends generalize to more complex disease conditions and, more broadly, to a wide variety of nonlinear dynamical systems.},
	urldate = {2022-11-22},
	publisher = {arXiv},
	author = {Linka, Kevin and Schafer, Amelie and Meng, Xuhui and Zou, Zongren and Karniadakis, George Em and Kuhl, Ellen},
	month = may,
	year = {2022},
	note = {arXiv:2205.08304 [nlin]},
	keywords = {62Mxx, 70Kxx, Computer Science - Machine Learning, G.3, J.3, Mathematics - Dynamical Systems, Nonlinear Sciences - Chaotic Dynamics, To Read, notion, unread},
}

@article{yang_b-pinns_2020,
	title = {B-{PINNs}: {Bayesian} {Physics}-{Informed} {Neural} {Networks} for {Forward} and {Inverse} {PDE} {Problems} with {Noisy} {Data}},
	url = {http://arxiv.org/abs/2003.06097},
	doi = {10.1016/j.jcp.2020.109913},
	abstract = {We propose a Bayesian physics-informed neural network (B-PINN) to solve both forward and inverse nonlinear problems described by partial differential equations (PDEs) and noisy data. In this Bayesian framework, the Bayesian neural network (BNN) combined with a PINN for PDEs serves as the prior while the Hamiltonian Monte Carlo (HMC) or the variational inference (VI) could serve as an estimator of the posterior. B-PINNs make use of both physical laws and scattered noisy measurements to provide predictions and quantify the aleatoric uncertainty arising from the noisy data in the Bayesian framework. Compared with PINNs, in addition to uncertainty quantification, B-PINNs obtain more accurate predictions in scenarios with large noise due to their capability of avoiding overfitting. We conduct a systematic comparison between the two different approaches for the B-PINN posterior estimation (i.e., HMC or VI), along with dropout used for quantifying uncertainty in deep neural networks. Our experiments show that HMC is more suitable than VI for the B-PINNs posterior estimation, while dropout employed in PINNs can hardly provide accurate predictions with reasonable uncertainty. Finally, we replace the BNN in the prior with a truncated Karhunen-Lo{\textbackslash}`eve (KL) expansion combined with HMC or a deep normalizing flow (DNF) model as posterior estimators. The KL is as accurate as BNN and much faster but this framework cannot be easily extended to high-dimensional problems unlike the BNN based framework.},
	author = {Yang, Liu and Meng, Xuhui and Karniadakis, George Em},
	month = mar,
	year = {2020},
	note = {arXiv: 2003.06097},
	keywords = {DEs, notion, unread},
}

@article{huang_applications_2023,
	title = {Applications of {Physics}-{Informed} {Neural} {Networks} in {Power} {Systems} - {A} {Review}},
	volume = {38},
	issn = {1558-0679},
	url = {https://ieeexplore.ieee.org/abstract/document/9743327?casa_token=6qlipmPVAKMAAAAA:2wpHuzKGE0CrIamcifalKJQ7eojIurtWpbdHBpCOmbrQW7zCexSCl49Z0-Pkik1UpD7A05K_ZA},
	doi = {10.1109/TPWRS.2022.3162473},
	abstract = {The advances of deep learning (DL) techniques bring new opportunities to numerous intractable tasks in power systems (PSs). Nevertheless, the extension of the application of DL in the domain of PSs has encountered challenges, e.g., high requirement for the quality and quantity of training data, production of physically infeasible/inconsistent solutions, and low generalizability and interpretability. There is a growing consensus that physics-informed neural networks (PINNs) can address these concerns by integrating physics-informed (PI) rules or laws into state-of-the-art DL methodology. This survey presents a systematic overview of the PINN in the domain of PSs. Specifically, several paradigms of PINN (e.g., PI loss function, PI initialization, PI design of architecture, and hybrid physics-DL models) are summarized. The applications of PINN in PSs in recent years, including state/parameter estimation, dynamic analysis, power flow calculation, optimal power flow, anomaly detection and location, and model and data synthesis, etc., are investigated in detail, followed by the summary and assessment of relevant works so far. Revolving around the characteristics of PSs and the state-of-the-art DL techniques, this paper outlines the potential research directions and attempts to shed light on the deeper and broader application of PINN on PSs.},
	number = {1},
	urldate = {2023-10-06},
	journal = {IEEE Transactions on Power Systems},
	author = {Huang, Bin and Wang, Jianhui},
	month = jan,
	year = {2023},
	note = {Conference Name: IEEE Transactions on Power Systems},
	keywords = {unread},
	pages = {572--588},
}

@book{tronstad_physics-informed_2022,
	title = {A {Physics}-{Informed} {Deep} {Learning} {Framework} for {Solving} {Inverse} {Problems} in {Epidemiology}},
	url = {http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-321233},
	abstract = {DiVA portal is a finding tool for research publications and student theses written at the following 50 universities and research institutions.},
	language = {eng},
	urldate = {2022-11-22},
	author = {Tronstad, Magnus},
	year = {2022},
	keywords = {unread},
}

@article{zhang_analyses_2022,
	title = {Analyses of internal structures and defects in materials using physics-informed neural networks},
	volume = {8},
	url = {https://www.science.org/doi/full/10.1126/sciadv.abk0644},
	doi = {10.1126/sciadv.abk0644},
	abstract = {Characterizing internal structures and defects in materials is a challenging task, often requiring solutions to inverse problems with unknown topology, geometry, material properties, and nonlinear deformation. Here, we present a general framework based on physics-informed neural networks for identifying unknown geometric and material parameters. By using a mesh-free method, we parameterize the geometry of the material using a differentiable and trainable method that can identify multiple structural features. We validate this approach for materials with internal voids/inclusions using constitutive models that encompass the spectrum of linear elasticity, hyperelasticity, and plasticity. We predict the size, shape, and location of the internal void/inclusion as well as the elastic modulus of the inclusion. Our general framework can be applied to other inverse problems in different applications that involve unknown material properties and highly deformable geometries, targeting material characterization, quality assurance, and structural design.},
	number = {7},
	urldate = {2022-11-21},
	journal = {Science Advances},
	author = {Zhang, Enrui and Dao, Ming and Karniadakis, George Em and Suresh, Subra},
	month = feb,
	year = {2022},
	note = {Publisher: American Association for the Advancement of Science},
	keywords = {unread},
	pages = {eabk0644},
}

@article{berkhahn_physics-informed_2022,
	title = {A physics-informed neural network to model {COVID}-19 infection and hospitalization scenarios},
	volume = {2022},
	issn = {2731-4235},
	url = {https://doi.org/10.1186/s13662-022-03733-5},
	doi = {10.1186/s13662-022-03733-5},
	abstract = {In this paper, we replace the standard numerical approach of estimating parameters in a mathematical model using numerical solvers for differential equations with a physics-informed neural network (PINN). This neural network requires a sequence of time instances as direct input of the network and the numbers of susceptibles, vaccinated, infected, hospitalized, and recovered individuals per time instance to learn certain parameters of the underlying model, which are used for the loss calculations.},
	language = {en},
	number = {1},
	urldate = {2022-11-22},
	journal = {Advances in Continuous and Discrete Models},
	author = {Berkhahn, Sarah and Ehrhardt, Matthias},
	month = oct,
	year = {2022},
	keywords = {COVID-19, Compartment models, Epidemiology, Physics-informed neural networks, SARS-CoV-2, SIR, unread},
	pages = {61},
}

@article{immer_effective_nodate,
	title = {Effective {Bayesian} {Heteroscedastic} {Regression} with {Deep} {Neural} {Networks}},
	abstract = {Flexibly quantifying both irreducible aleatoric and model-dependent epistemic uncertainties plays an important role for complex regression problems. While deep neural networks in principle can provide this flexibility and learn heteroscedastic aleatoric uncertainties through non-linear functions, recent works highlight that maximizing the log likelihood objective parameterized by mean and variance can lead to compromised mean fits since the gradient are scaled by the predictive variance, and propose adjustments in line with this premise. We instead propose to use the natural parametrization of the Gaussian, which has been shown to be more stable for heteroscedastic regression based on non-linear feature maps and Gaussian processes. Further, we emphasize the significance of principled regularization of the network parameters and prediction. We therefore propose an efficient Laplace approximation for heteroscedastic neural networks that allows automatic regularization through empirical Bayes and provides epistemic uncertainties, both of which improve generalization. We showcase on a range of regression problems—including a new heteroscedastic image regression benchmark—that our methods are scalable, improve over previous approaches for heteroscedastic regression, and provide epistemic uncertainty without requiring hyperparameter tuning.},
	language = {en},
	author = {Immer, Alexander and Palumbo, Emanuele and Marx, Alexander and Vogt, Julia E},
	keywords = {unread},
}

@book{wong-toi_understanding_2023,
	title = {Understanding {Pathologies} of {Deep} {Heteroskedastic} {Regression}},
	abstract = {Several recent studies have reported negative results when using heteroskedastic neural regression models to model real-world data. In particular, for overparameterized models, the mean and variance networks are powerful enough to either fit every single data point (while shrinking the predicted variances to zero), or to learn a constant prediction with an output variance exactly matching every predicted residual (i.e., explaining the targets as pure noise). This paper studies these difficulties from the perspective of statistical physics. We show that the observed instabilities are not specific to any neural network architecture but are already present in a field theory of an overparameterized conditional Gaussian likelihood model. Under light assumptions, we derive a nonparametric free energy that can be solved numerically. The resulting solutions show excellent qualitative agreement with empirical model fits on real-world data and, in particular, prove the existence of phase transitions, i.e., abrupt, qualitative differences in the behaviors of the regressors upon varying the regularization strengths on the two networks. Our work thus provides a theoretical explanation for the necessity to carefully regularize heteroskedastic regression models. Moreover, the insights from our theory suggest a scheme for optimizing this regularization which is quadratically more efficient than the naive approach.},
	author = {Wong-Toi, Eliot and Boyd, Alex and Fortuin, Vincent and Mandt, Stephan},
	month = jun,
	year = {2023},
	keywords = {unread},
}

@article{seitzer_pitfalls_2022,
	title = {{ON} {THE} {PITFALLS} {OF} {HETEROSCEDASTIC} {UNCERTAINTY} {ESTIMATION} {WITH} {PROBABILISTIC} {NEURAL} {NETWORKS}},
	abstract = {Capturing aleatoric uncertainty is a critical part of many machine learning systems. In deep learning, a common approach to this end is to train a neural network to estimate the parameters of a heteroscedastic Gaussian distribution by maximizing the logarithm of the likelihood function under the observed data. In this work, we examine this approach and identify potential hazards associated with the use of log-likelihood in conjunction with gradient-based optimizers. First, we present a synthetic example illustrating how this approach can lead to very poor but stable parameter estimates. Second, we identify the culprit to be the log-likelihood loss, along with certain conditions that exacerbate the issue. Third, we present an alternative formulation, termed β−NLL, in which each data point’s contribution to the loss is weighted by the β-exponentiated variance estimate. We show that using an appropriate β largely mitigates the issue in our illustrative example. Fourth, we evaluate this approach on a range of domains and tasks and show that it achieves considerable improvements and performs more robustly concerning hyperparameters, both in predictive RMSE and log-likelihood criteria.},
	language = {en},
	author = {Seitzer, Maximilian and Tavakoli, Arash and Antic, Dimitrije and Martius, Georg},
	year = {2022},
	keywords = {unread},
}

@misc{schweighofer_introducing_2023,
	title = {Introducing an {Improved} {Information}-{Theoretic} {Measure} of {Predictive} {Uncertainty}},
	url = {http://arxiv.org/abs/2311.08309},
	abstract = {Applying a machine learning model for decision-making in the real world requires to distinguish what the model knows from what it does not. A critical factor in assessing the knowledge of a model is to quantify its predictive uncertainty. Predictive uncertainty is commonly measured by the entropy of the Bayesian model average (BMA) predictive distribution. Yet, the properness of this current measure of predictive uncertainty was recently questioned. We provide new insights regarding those limitations. Our analyses show that the current measure erroneously assumes that the BMA predictive distribution is equivalent to the predictive distribution of the true model that generated the dataset. Consequently, we introduce a theoretically grounded measure to overcome these limitations. We experimentally verify the benefits of our introduced measure of predictive uncertainty. We find that our introduced measure behaves more reasonably in controlled synthetic tasks. Moreover, our evaluations on ImageNet demonstrate that our introduced measure is advantageous in real-world applications utilizing predictive uncertainty.},
	urldate = {2023-11-21},
	publisher = {arXiv},
	author = {Schweighofer, Kajetan and Aichberger, Lukas and Ielanskyi, Mykyta and Hochreiter, Sepp},
	month = nov,
	year = {2023},
	note = {arXiv:2311.08309 [cs, stat]},
	keywords = {Computer Science - Machine Learning, Statistics - Machine Learning, unread},
}

@misc{chantada_cosmological_2022,
	title = {Cosmological informed neural networks to solve the background dynamics of the {Universe}},
	url = {http://arxiv.org/abs/2205.02945},
	abstract = {The field of machine learning has drawn increasing interest from various other fields due to the success of its methods at solving a plethora of different problems. An application of these has been to train artificial neural networks to solve differential equations without the need of a numerical solver. This particular application offers an alternative to conventional numerical methods, with advantages such as lower memory required to store the solutions, parallelization, and in some cases less overall computational cost than its numerical counterparts. In this work, we train artificial neural networks to represent a bundle of solutions of the differential equations that govern the background dynamics of the Universe for four different models. The models we have chosen are \${\textbackslash}Lambda {\textbackslash}mathrm\{CDM\}\$, the Chevallier-Polarski-Linder parametric dark energy model, a quintessence model with an exponential potential, and the Hu-Sawicki \$f{\textbackslash}left(R{\textbackslash}right)\$ model. We used the solutions that the networks provide to perform statistical analyses to estimate the values of each model's parameters with observational data; namely, estimates of the Hubble parameter from Cosmic Chronometers, the Supernovae type Ia data from the Pantheon compilation, and measurements from Baryon Acoustic Oscillations. The results we obtain for all models match similar estimations done in the literature using numerical solvers. In addition, we estimated the error of the solutions by comparing them to the analytical solution when there is one, or to a high-precision numerical solution when there is not. Through those estimations we found that the error of the solutions was at most \${\textbackslash}sim1{\textbackslash}\%\$ in the region of the parameter space that concerns the \$95{\textbackslash}\%\$ confidence regions that we found using the data, for all models and all statistical analyses performed in this work.},
	urldate = {2022-11-08},
	publisher = {arXiv},
	author = {Chantada, Augusto T. and Landau, Susana J. and Protopapas, Pavlos and Scóccola, Claudia G. and Garraffo, Cecilia},
	month = may,
	year = {2022},
	note = {arXiv:2205.02945 [astro-ph, physics:gr-qc, physics:hep-ph]},
	keywords = {Astrophysics - Cosmology and Nongalactic Astrophysics, General Relativity and Quantum Cosmology, High Energy Physics - Phenomenology, Read, notion},
}

@article{zhang_cong_four_2014,
	title = {Four new observational {H}(z) data from luminous red galaxies in the {Sloan} {Digital} {Sky} {Survey} data release seven},
	volume = {14},
	issn = {1674-4527},
	url = {https://dx.doi.org/10.1088/1674-4527/14/10/002},
	doi = {10.1088/1674-4527/14/10/002},
	abstract = {By adopting the differential age method, we select 17 832 luminous red galaxies from the Sloan Digital Sky Survey Data Release Seven covering redshift 0 {\textless} z {\textless} 0.4 to measure the Hubble parameter. Using the full spectrum fitting package UlySS, these spectra are reduced with single stellar population models and optimal age information from our selected sample is derived. With the decreasing age-redshift relation, four new observational H(z) data (OHD) points are obtained, which are H(z) = 69.0 ± 19.6 km s−1 Mpc−1 at z = 0.07, H(z) = 68.6 ± 26.2km s−1 Mpc−1 at z = 0.12, H(z) = 72.9 ± 29.6 km s−1 Mpc−1 at z = 0.2 and H(z)=88.8 ± 36.6 km s−1 Mpc−1 at z = 0.28, respectively. Combined with 21 other available OHD data points, the performance of the constraint on both flat and non-flat ΛCDM models is presented.},
	language = {en},
	number = {10},
	urldate = {2023-10-01},
	journal = {Research in Astronomy and Astrophysics},
	author = {Zhang Cong and Zhang Han and Yuan Shuo and Liu Siqi and Zhang Tong-Jie and Sun Yan-Chun},
	month = oct,
	year = {2014},
	keywords = {unread},
	pages = {1221},
}

@article{moresco_raising_2015,
	title = {Raising the bar: new constraints on the {Hubble} parameter with cosmic chronometers at z  2},
	volume = {450},
	issn = {1745-3925},
	shorttitle = {Raising the bar},
	url = {https://doi.org/10.1093/mnrasl/slv037},
	doi = {10.1093/mnrasl/slv037},
	abstract = {One of the most compelling tasks of modern cosmology is to constrain the expansion history of the Universe, since this measurement can give insights on the nature of dark energy and help to estimate cosmological parameters. In this letter are presented two new measurements of the Hubble parameter H(z) obtained with the cosmic chronometer method up to z  2. Taking advantage of near-infrared spectroscopy of the few very massive and passive galaxies observed at z \&gt; 1.4 available in literature, the differential evolution of this population is estimated and calibrated with different stellar population synthesis models to constrain H(z), including in the final error budget all possible sources of systematic uncertainties (star formation history, stellar metallicity, model dependences). This analysis is able to extend significantly the redshift range coverage with respect to present-day constraints, crossing for the first time the limit at z  1.75. The new H(z) data are used to estimate the gain in accuracy on cosmological parameters with respect to previous measurements in two cosmological models, finding a small but detectable improvement (5 per cent) in particular on ΩM and w0. Finally, a simulation of a Euclid-like survey has been performed to forecast the expected improvement with future data. The provided constraints have been obtained just with the cosmic chronometers approach, without any additional data, and the results show the high potentiality of this method to constrain the expansion history of the Universe at these redshifts.},
	number = {1},
	urldate = {2023-10-01},
	journal = {Monthly Notices of the Royal Astronomical Society: Letters},
	author = {Moresco, Michele},
	month = jun,
	year = {2015},
	keywords = {unread},
	pages = {L16--L20},
}

@article{moresco_improved_2012,
	title = {Improved constraints on the expansion rate of the {Universe} up to z  1.1 from the spectroscopic evolution of cosmic chronometers},
	volume = {2012},
	issn = {1475-7516},
	url = {https://dx.doi.org/10.1088/1475-7516/2012/08/006},
	doi = {10.1088/1475-7516/2012/08/006},
	abstract = {We present new improved constraints on the Hubble parameter H(z) in the redshift range 0.15 {\textless} z {\textless} 1.1, obtained from the differential spectroscopic evolution of early-type galaxies as a function of redshift. We extract a large sample of early-type galaxies ( ∼ 11000) from several spectroscopic surveys, spanning almost 8 billion years of cosmic lookback time (0.15 {\textless} z {\textless} 1.42). We select the most massive, red elliptical galaxies, passively evolving and without signature of ongoing star formation. Those galaxies can be used as standard cosmic chronometers, as firstly proposed by Jimenez \& Loeb (2002), whose differential age evolution as a function of cosmic time directly probes H(z). We analyze the 4000 Å break (D4000) as a function of redshift, use stellar population synthesis models to theoretically calibrate the dependence of the differential age evolution on the differential D4000, and estimate the Hubble parameter taking into account both statistical and systematical errors. We provide 8 new measurements of H(z) (see table 4), and determine its change in H(z) to a precision of 5–12\% mapping homogeneously the redshift range up to z ∼ 1.1; for the first time, we place a constraint on H(z) at z≠0 with a precision comparable with the one achieved for the Hubble constant (about 5–6\% at z ∼ 0.2), and covered a redshift range (0.5 {\textless} z {\textless} 0.8) which is crucial to distinguish many different quintessence cosmologies. These measurements have been tested to best match a ΛCDM model, clearly providing a statistically robust indication that the Universe is undergoing an accelerated expansion. This method shows the potentiality to open a new avenue in constrain a variety of alternative cosmologies, especially when future surveys (e.g. Euclid) will open the possibility to extend it up to z ∼ 2.},
	language = {en},
	number = {08},
	urldate = {2023-10-01},
	journal = {Journal of Cosmology and Astroparticle Physics},
	author = {Moresco, M. and Cimatti, A. and Jimenez, R. and Pozzetti, L. and Zamorani, G. and Bolzonella, M. and Dunlop, J. and Lamareille, F. and Mignoli, M. and Pearce, H. and Rosati, P. and Stern, D. and Verde, L. and Zucca, E. and Carollo, C. M. and Contini, T. and Kneib, J.-P. and Fèvre, O. Le and Lilly, S. J. and Mainieri, V. and Renzini, A. and Scodeggio, M. and Balestra, I. and Gobat, R. and McLure, R. and Bardelli, S. and Bongiorno, A. and Caputi, K. and Cucciati, O. and Torre, S. de la and Ravel, L. de and Franzetti, P. and Garilli, B. and Iovino, A. and Kampczyk, P. and Knobel, C. and Kovač, K. and Borgne, J.-F. Le and Brun, V. Le and Maier, C. and Pelló, R. and Peng, Y. and Perez-Montero, E. and Presotto, V. and Silverman, J. D. and Tanaka, M. and Tasca, L. A. M. and Tresse, L. and Vergani, D. and Almaini, O. and Barnes, L. and Bordoloi, R. and Bradshaw, E. and Cappi, A. and Chuter, R. and Cirasuolo, M. and Coppa, G. and Diener, C. and Foucaud, S. and Hartley, W. and Kamionkowski, M. and Koekemoer, A. M. and López-Sanjuan, C. and McCracken, H. J. and Nair, P. and Oesch, P. and Stanford, A. and Welikala, N.},
	month = aug,
	year = {2012},
	keywords = {unread},
	pages = {006},
}

@article{moresco_6_2016,
	title = {A 6\% measurement of the {Hubble} parameter at z 0.45: direct evidence of the epoch of cosmic re-acceleration},
	volume = {2016},
	issn = {1475-7516},
	shorttitle = {A 6\% measurement of the {Hubble} parameter at z∼0.45},
	url = {https://dx.doi.org/10.1088/1475-7516/2016/05/014},
	doi = {10.1088/1475-7516/2016/05/014},
	abstract = {Deriving the expansion history of the Universe is a major goal of modern cosmology. To date, the most accurate measurements have been obtained with Type Ia Supernovae (SNe) and Baryon Acoustic Oscillations (BAO), providing evidence for the existence of a transition epoch at which the expansion rate changes from decelerated to accelerated. However, these results have been obtained within the framework of specific cosmological models that must be implicitly or explicitly assumed in the measurement. It is therefore crucial to obtain measurements of the accelerated expansion of the Universe independently of assumptions on cosmological models. Here we exploit the unprecedented statistics provided by the Baryon Oscillation Spectroscopic Survey (BOSS, [1-3]) Data Release 9 to provide new constraints on the Hubble parameter H(z) using the cosmic chronometers approach. We extract a sample of more than 130000 of the most massive and passively evolving galaxies, obtaining five new cosmology-independent H(z) measurements in the redshift range 0.3 {\textless} z {\textless} 0.5, with an accuracy of ∼11–16\% incorporating both statistical and systematic errors. Once combined, these measurements yield a 6\% accuracy constraint of H(z = 0.4293) = 91.8 ± 5.3 km/s/Mpc. The new data are crucial to provide the first cosmology-independent determination of the transition redshift at high statistical significance, measuring zt = 0.4 ± 0.1, and to significantly disfavor the null hypothesis of no transition between decelerated and accelerated expansion at 99.9\% confidence level. This analysis highlights the wide potential of the cosmic chronometers approach: it permits to derive constraints on the expansion history of the Universe with results competitive with standard probes, and most importantly, being the estimates independent of the cosmological model, it can constrain cosmologies beyond—and including—the ΛCDM model.},
	language = {en},
	number = {05},
	urldate = {2023-10-01},
	journal = {Journal of Cosmology and Astroparticle Physics},
	author = {Moresco, Michele and Pozzetti, Lucia and Cimatti, Andrea and Jimenez, Raul and Maraston, Claudia and Verde, Licia and Thomas, Daniel and Citro, Annalisa and Tojeiro, Rita and Wilkinson, David},
	month = may,
	year = {2016},
	keywords = {unread},
	pages = {014},
}

@article{simon_constraints_2005,
	title = {Constraints on the redshift dependence of the dark energy potential},
	volume = {71},
	url = {https://link.aps.org/doi/10.1103/PhysRevD.71.123001},
	doi = {10.1103/PhysRevD.71.123001},
	abstract = {We develop a formalism to characterize the shape and the redshift evolution of the dark energy potential. Our formalism makes use of quantities similar to the horizon-flow parameters in inflation and is general enough that can deal with multiscalar quintessence scenarios, exotic matter components, and higher-order curvature corrections to General Relativity. We show how the shape of the dark energy potential can be recovered nonparametrically using this formalism and we present approximations analogous to the ones relevant to slow-roll inflation. Since presently available data do not allow a nonparametric and exact reconstruction of the potential, we consider a general parametric description. This reconstruction can also be used in other approaches followed in the literature (e.g., the reconstruction of the redshift evolution of the dark energy equation of state w(z)). Using observations of passively evolving galaxies and supernova data we derive constraints on the dark energy potential shape in the redshift range 0.1{\textless}z{\textless}1.8. Our findings show that at the 1σ level the potential is consistent with being constant, although at the same level of confidence variations cannot be excluded with current data. We forecast constraints achievable with future data from the Atacama Cosmology Telescope.},
	number = {12},
	urldate = {2023-10-01},
	journal = {Physical Review D},
	author = {Simon, Joan and Verde, Licia and Jimenez, Raul},
	month = jun,
	year = {2005},
	note = {Publisher: American Physical Society},
	keywords = {unread},
	pages = {123001},
}

@article{copeland_exponential_1998,
	title = {Exponential potentials and cosmological scaling solutions},
	volume = {57},
	url = {https://link.aps.org/doi/10.1103/PhysRevD.57.4686},
	doi = {10.1103/PhysRevD.57.4686},
	abstract = {We present a phase-plane analysis of cosmologies containing a baryotropic fluid with an equation of state pγ=(γ−1)ργ, plus a scalar field φ with an exponential potential V∝exp(−λκφ) where κ2=8πG. In addition to the well-known inflationary solutions for λ2{\textless}2, there exist scaling solutions when λ2{\textgreater}3γ in which the scalar field energy density tracks that of the baryotropic fluid (which for example might be radiation or dust). We show that the scaling solutions are the unique late-time attractors whenever they exist. The fluid-dominated solutions, where V(φ)/ργ→0 at late times, are always unstable (except for the cosmological constant case γ=0). The relative energy density of the fluid and scalar field depends on the steepness of the exponential potential, which is constrained by nucleosynthesis to λ2{\textgreater}20. We show that standard inflation models are unable to solve this “relic density” problem.},
	number = {8},
	urldate = {2023-10-01},
	journal = {Physical Review D},
	author = {Copeland, Edmund J. and Liddle, Andrew R. and Wands, David},
	month = apr,
	year = {1998},
	note = {Publisher: American Physical Society},
	keywords = {unread},
	pages = {4686--4690},
}

@article{armendariz-picon_dynamical_2000,
	title = {Dynamical {Solution} to the {Problem} of a {Small} {Cosmological} {Constant} and {Late}-{Time} {Cosmic} {Acceleration}},
	volume = {85},
	url = {https://link.aps.org/doi/10.1103/PhysRevLett.85.4438},
	doi = {10.1103/PhysRevLett.85.4438},
	abstract = {Increasing evidence suggests that most of the energy density of the universe consists of a dark energy component with negative pressure that causes the cosmic expansion to accelerate. We address why this component comes to dominate the universe only recently. We present a class of theories based on an evolving scalar field where the explanation is based entirely on internal dynamical properties of the solutions. In the theories we consider, the dynamics causes the scalar field to lock automatically into a negative pressure state at the onset of matter domination such that the present epoch is the earliest possible time consistent with nucleosynthesis restrictions when it can start to dominate.},
	number = {21},
	urldate = {2023-10-01},
	journal = {Physical Review Letters},
	author = {Armendariz-Picon, C. and Mukhanov, V. and Steinhardt, Paul J.},
	month = nov,
	year = {2000},
	note = {Publisher: American Physical Society},
	keywords = {unread},
	pages = {4438--4441},
}

@article{zlatev_quintessence_1999,
	title = {Quintessence, {Cosmic} {Coincidence}, and the {Cosmological} {Constant}},
	volume = {82},
	url = {https://link.aps.org/doi/10.1103/PhysRevLett.82.896},
	doi = {10.1103/PhysRevLett.82.896},
	abstract = {Recent observations suggest that a large fraction of the energy density of the Universe has negative pressure. One explanation is vacuum energy density; another is quintessence in the form of a scalar field slowly evolving down a potential. In either case, a key problem is to explain why the energy density nearly coincides with the matter density today. The densities decrease at different rates as the Universe expands, so coincidence today appears to require that their ratio be set to a specific, infinitesimal value in the early Universe. In this paper, we introduce the notion of a “tracker field,” a form of quintessence, and show how it may explain the coincidence, adding new motivation for the quintessence scenario.},
	number = {5},
	urldate = {2023-10-01},
	journal = {Physical Review Letters},
	author = {Zlatev, Ivaylo and Wang, Limin and Steinhardt, Paul J.},
	month = feb,
	year = {1999},
	note = {Publisher: American Physical Society},
	keywords = {unread},
	pages = {896--899},
}

@article{linder_exploring_2003,
	title = {Exploring the {Expansion} {History} of the {Universe}},
	volume = {90},
	url = {https://link.aps.org/doi/10.1103/PhysRevLett.90.091301},
	doi = {10.1103/PhysRevLett.90.091301},
	abstract = {Exploring the recent expansion history of the universe promises insights into the cosmological model, the nature of dark energy, and potentially clues to high energy physics theories and gravitation. We examine the extent to which precision distance-redshift observations can map out the history, including the acceleration-deceleration transition, and the components and equations of state of the energy density. We consider the ability to distinguish between various dynamical scalar field models for the dark energy, as well as higher dimension and alternate gravity theories. Finally, we present a new, advantageous parametrization for the study of dark energy.},
	number = {9},
	urldate = {2023-10-01},
	journal = {Physical Review Letters},
	author = {Linder, Eric V.},
	month = mar,
	year = {2003},
	note = {Publisher: American Physical Society},
	keywords = {unread},
	pages = {091301},
}

@article{stern_cosmic_2010,
	title = {Cosmic chronometers: constraining the equation of state of dark energy. {I}: {H}(z) measurements},
	volume = {2010},
	issn = {1475-7516},
	shorttitle = {Cosmic chronometers},
	url = {https://dx.doi.org/10.1088/1475-7516/2010/02/008},
	doi = {10.1088/1475-7516/2010/02/008},
	abstract = {We present new determinations of the cosmic expansion history from red-envelope galaxies. We have obtained for this purpose high-quality spectra with the Keck-LRIS spectrograph of red-envelope galaxies in 24 galaxy clusters in the redshift range 0.2 {\textless} z {\textless} 1.0. We complement these Keck spectra with high-quality, publicly available archival spectra from the SPICES and VVDS surveys. We improve over our previous expansion history measurements in Simon et al. (2005) by providing two new determinations of the expansion history: H(z) = 97±62 km sec−1 Mpc−1 at z ≃ 0.5 and H(z) = 90±40 km sec−1 Mpc−1 at z ≃ 0.9. We discuss the uncertainty in the expansion history determination that arises from uncertainties in the synthetic stellar-population models. We then use these new measurements in concert with cosmic-microwave-background (CMB) measurements to constrain cosmological parameters, with a special emphasis on dark-energy parameters and constraints to the curvature. In particular, we demonstrate the usefulness of direct H(z) measurements by constraining the dark-energy equation of state parameterized by w0 and wa and allowing for arbitrary curvature. Further, we also constrain, using only CMB and H(z) data, the number of relativistic degrees of freedom to be 4±0.5 and their total mass to be {\textless} 0.2 eV, both at 1σ.},
	language = {en},
	number = {02},
	urldate = {2023-10-01},
	journal = {Journal of Cosmology and Astroparticle Physics},
	author = {Stern, Daniel and Jimenez, Raul and Verde, Licia and Kamionkowski, Marc and Stanford, S. Adam},
	month = feb,
	year = {2010},
	keywords = {unread},
	pages = {008},
}

@article{caldwell_cosmological_1998,
	title = {Cosmological {Imprint} of an {Energy} {Component} with {General} {Equation} of {State}},
	volume = {80},
	url = {https://link.aps.org/doi/10.1103/PhysRevLett.80.1582},
	doi = {10.1103/PhysRevLett.80.1582},
	abstract = {We examine the possibility that a significant component of the energy density of the Universe has an equation of state different from that of matter, radiation, or cosmological constant ( Λ). An example is a cosmic scalar field evolving in a potential, but our treatment is more general. Including this component alters cosmic evolution in a way that fits current observations well. Unlike Λ, it evolves dynamically and develops fluctuations, leaving a distinctive imprint on the microwave background anisotropy and mass power spectrum.},
	number = {8},
	urldate = {2023-10-01},
	journal = {Physical Review Letters},
	author = {Caldwell, R. R. and Dave, Rahul and Steinhardt, Paul J.},
	month = feb,
	year = {1998},
	note = {Publisher: American Physical Society},
	keywords = {unread},
	pages = {1582--1585},
}

@article{chevallier_accelerating_2001,
	title = {Accelerating universes with scaling dark matter},
	volume = {10},
	issn = {0218-2718},
	url = {https://www.worldscientific.com/doi/abs/10.1142/S0218271801000822},
	doi = {10.1142/S0218271801000822},
	abstract = {Friedmann–Robertson–Walker universes with a presently large fraction of the energy density stored in an X-component with wX{\textless}-1/3, are considered. We find all the critical points of the system for constant equations of state in that range. We consider further several background quantities that can distinguish the models with different wX values. Using a simple toy model with a varying equation of state, we show that even a large variation of wX at small redshifts is very difficult to observe with dL(z) measurements up to z{\textasciitilde}1. Therefore, it will require accurate measurements in the range 1{\textless}z{\textless}2 and independent accurate knowledge of Ωm,0 (and/or ΩX,0) in order to resolve a variable wX from a constant wX.},
	number = {02},
	urldate = {2023-10-01},
	journal = {International Journal of Modern Physics D},
	author = {Chevallier, Michel and Polarski, David},
	month = apr,
	year = {2001},
	note = {Publisher: World Scientific Publishing Co.},
	keywords = {unread},
	pages = {213--223},
}

@article{buchdahl_non-linear_1970,
	title = {Non-{Linear} {Lagrangians} and {Cosmological} {Theory}},
	volume = {150},
	issn = {0035-8711},
	url = {https://doi.org/10.1093/mnras/150.1.1},
	doi = {10.1093/mnras/150.1.1},
	abstract = {In relativistic cosmology the theory of uniform model universes is based on Einstein's equations, which derive from a variational principle the field-Lagrangian L of which is the scalar curvature R to within an arbitrary additive constant. In this work the possibility of taking L to be a more general invariant of the Riemann tensor is contemplated. The consequences of choosing L to be a function ϕ of R alone are tentatively examined under specialized circumstances, with particular attention to an open world-model oscillating between non-singular states. Difficulties revolving about the actual form which ϕ might take are discussed.},
	number = {1},
	urldate = {2023-10-01},
	journal = {Monthly Notices of the Royal Astronomical Society},
	author = {Buchdahl, H. A.},
	month = sep,
	year = {1970},
	keywords = {unread},
	pages = {1--8},
}

@article{clifton_modified_2012,
	series = {Modified {Gravity} and {Cosmology}},
	title = {Modified gravity and cosmology},
	volume = {513},
	issn = {0370-1573},
	url = {https://www.sciencedirect.com/science/article/pii/S0370157312000105},
	doi = {10.1016/j.physrep.2012.01.001},
	abstract = {In this review we present a thoroughly comprehensive survey of recent work on modified theories of gravity and their cosmological consequences. Amongst other things, we cover General Relativity, scalar–tensor, Einstein–æther, and Bimetric theories, as well as TeVeS, f(R), general higher-order theories, Hořava–Lifschitz gravity, Galileons, Ghost Condensates, and models of extra dimensions including Kaluza–Klein, Randall–Sundrum, DGP, and higher co-dimension braneworlds. We also review attempts to construct a Parameterised Post-Friedmannian formalism, that can be used to constrain deviations from General Relativity in cosmology, and that is suitable for comparison with data on the largest scales. These subjects have been intensively studied over the past decade, largely motivated by rapid progress in the field of observational cosmology that now allows, for the first time, precision tests of fundamental physics on the scale of the observable Universe. The purpose of this review is to provide a reference tool for researchers and students in cosmology and gravitational physics, as well as a self-contained, comprehensive and up-to-date introduction to the subject as a whole.},
	number = {1},
	urldate = {2023-10-01},
	journal = {Physics Reports},
	author = {Clifton, Timothy and Ferreira, Pedro G. and Padilla, Antonio and Skordis, Constantinos},
	month = mar,
	year = {2012},
	keywords = {Cosmology, General relativity, Gravitational physics, Modified gravity, unread},
	pages = {1--189},
}

@article{talavera_differential_nodate-1,
	title = {Differential equations system to describe bacterial growth, {pH} variation and lactic acid production in batch fermentation},
	language = {en},
	author = {Talavera, Eng Emmanuel Rodríguez and Antonio, Dr Paul and Trujillo, Valle and Muñoz, Dra Yolocuauhtli Salazar},
	keywords = {Read, notion},
}

@article{moreno-zambrano_mathematical_2018-1,
	title = {A mathematical model of cocoa bean fermentation},
	volume = {5},
	issn = {2054-5703},
	url = {https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6227950/},
	doi = {10.1098/rsos.180964},
	abstract = {Cocoa bean fermentation relies on the sequential activation of several microbial populations, triggering a temporal pattern of biochemical transformations. Understanding this complex process is of tremendous importance as it is known to form the precursors of the resulting chocolate’s flavour and taste. At the same time, cocoa bean fermentation is one of the least controlled processes in the food industry. Here, a quantitative model of cocoa bean fermentation is constructed based on available microbiological and biochemical knowledge. The model is formulated as a system of coupled ordinary differential equations with two distinct types of state variables: (i) metabolite concentrations of glucose, fructose, ethanol, lactic acid and acetic acid and (ii) population sizes of yeast, lactic acid bacteria and acetic acid bacteria. We demonstrate that the model can quantitatively describe existing fermentation time series and that the estimated parameters, obtained by a Bayesian framework, can be used to extract and interpret differences in environmental conditions. The proposed model is a valuable tool towards a mechanistic understanding of this complex biochemical process, and can serve as a starting point for hypothesis testing of new systemic adjustments. In addition to providing the first quantitative mathematical model of cocoa bean fermentation, the purpose of our investigation is to show how differences in estimated parameter values for two experiments allow us to deduce differences in experimental conditions.},
	number = {10},
	urldate = {2022-12-22},
	journal = {Royal Society Open Science},
	author = {Moreno-Zambrano, Mauricio and Grimbs, Sergio and Ullrich, Matthias S. and Hütt, Marc-Thorsten},
	month = oct,
	year = {2018},
	pmid = {30473841},
	pmcid = {PMC6227950},
	keywords = {Read, notion, unread},
	pages = {180964},
}

@article{kiureghian_aleatory_2009,
	series = {Risk {Acceptance} and {Risk} {Communication}},
	title = {Aleatory or epistemic? {Does} it matter?},
	volume = {31},
	issn = {0167-4730},
	shorttitle = {Aleatory or epistemic?},
	url = {https://www.sciencedirect.com/science/article/pii/S0167473008000556},
	doi = {10.1016/j.strusafe.2008.06.020},
	abstract = {The sources and characters of uncertainties in engineering modeling for risk and reliability analyses are discussed. While many sources of uncertainty may exist, they are generally categorized as either aleatory or epistemic. Uncertainties are characterized as epistemic, if the modeler sees a possibility to reduce them by gathering more data or by refining models. Uncertainties are categorized as aleatory if the modeler does not foresee the possibility of reducing them. From a pragmatic standpoint, it is useful to thus categorize the uncertainties within a model, since it then becomes clear as to which uncertainties have the potential of being reduced. More importantly, epistemic uncertainties may introduce dependence among random events, which may not be properly noted if the character of uncertainties is not correctly modeled. Influences of the two types of uncertainties in reliability assessment, codified design, performance-based engineering and risk-based decision-making are discussed. Two simple examples demonstrate the influence of statistical dependence arising from epistemic uncertainties on systems and time-variant reliability problems.},
	number = {2},
	urldate = {2023-10-19},
	journal = {Structural Safety},
	author = {Kiureghian, Armen Der and Ditlevsen, Ove},
	month = mar,
	year = {2009},
	keywords = {Aleatory, Epistemic, Ergodicity, Parameter uncertainty, Predictive models, Probability distribution choice, Read, Statistical dependence, Systems, Time-variant reliability, Uncertainty},
	pages = {105--112},
}

@article{zhang_basic_2020,
	title = {Basic {Framework} and {Main} {Methods} of {Uncertainty} {Quantification}},
	volume = {2020},
	issn = {1024-123X},
	url = {https://www.hindawi.com/journals/mpe/2020/6068203/},
	doi = {10.1155/2020/6068203},
	abstract = {Since 2000, the research of uncertainty quantification (UQ) has been successfully applied in many fields and has been highly valued and strongly supported by academia and industry. This review firstly discusses the sources and the types of uncertainties and gives an overall discussion on the goal, practical significance, and basic framework of the research of UQ. Then, the core ideas and typical methods of several important UQ processes are introduced, including sensitivity analysis, uncertainty propagation, model calibration, Bayesian inference, experimental design, surrogate model, and model uncertainty analysis.},
	language = {en},
	urldate = {2023-10-11},
	journal = {Mathematical Problems in Engineering},
	author = {Zhang, Juan and Yin, Junping and Wang, Ruili},
	month = aug,
	year = {2020},
	note = {Publisher: Hindawi},
	keywords = {Read},
	pages = {e6068203},
}

@article{caldeira_deeply_2020,
	title = {Deeply uncertain: comparing methods of uncertainty quantification in deep learning algorithms},
	volume = {2},
	issn = {2632-2153},
	shorttitle = {Deeply uncertain},
	url = {https://dx.doi.org/10.1088/2632-2153/aba6f3},
	doi = {10.1088/2632-2153/aba6f3},
	abstract = {We present a comparison of methods for uncertainty quantification (UQ) in deep learning algorithms in the context of a simple physical system. Three of the most common uncertainty quantification methods—Bayesian neural networks (BNNs), concrete dropout (CD), and deep ensembles (DEs) — are compared to the standard analytic error propagation. We discuss this comparison in terms endemic to both machine learning (‘epistemic’ and ‘aleatoric’) and the physical sciences (‘statistical’ and ‘systematic’). The comparisons are presented in terms of simulated experimental measurements of a single pendulum—a prototypical physical system for studying measurement and analysis techniques. Our results highlight some pitfalls that may occur when using these UQ methods. For example, when the variation of noise in the training set is small, all methods predicted the same relative uncertainty independently of the inputs. This issue is particularly hard to avoid in BNN. On the other hand, when the test set contains samples far from the training distribution, we found that no methods sufficiently increased the uncertainties associated to their predictions. This problem was particularly clear for CD. In light of these results, we make some recommendations for usage and interpretation of UQ methods.},
	language = {en},
	number = {1},
	urldate = {2023-10-11},
	journal = {Machine Learning: Science and Technology},
	author = {Caldeira, João and Nord, Brian},
	month = dec,
	year = {2020},
	note = {Publisher: IOP Publishing},
	keywords = {Read},
	pages = {015002},
}

@inproceedings{kendall_what_2017,
	title = {What {Uncertainties} {Do} {We} {Need} in {Bayesian} {Deep} {Learning} for {Computer} {Vision}?},
	volume = {30},
	url = {https://papers.nips.cc/paper_files/paper/2017/hash/2650d6089a6d640c5e85b2b88265dc2b-Abstract.html},
	abstract = {There are two major types of uncertainty one can model. Aleatoric uncertainty captures noise inherent in the observations. On the other hand, epistemic uncertainty accounts for uncertainty in the model - uncertainty which can be explained away given enough data. Traditionally it has been difficult to model epistemic uncertainty in computer vision, but with new Bayesian deep learning tools this is now possible. We study the benefits of modeling epistemic vs. aleatoric uncertainty in Bayesian deep learning models for vision tasks. For this we present a Bayesian deep learning framework combining input-dependent aleatoric uncertainty together with epistemic uncertainty. We study models under the framework with per-pixel semantic segmentation and depth regression tasks. Further, our explicit uncertainty formulation leads to new loss functions for these tasks, which can be interpreted as learned attenuation. This makes the loss more robust to noisy data, also giving new state-of-the-art results on segmentation and depth regression benchmarks.},
	urldate = {2023-10-17},
	booktitle = {Advances in {Neural} {Information} {Processing} {Systems}},
	publisher = {Curran Associates, Inc.},
	author = {Kendall, Alex and Gal, Yarin},
	year = {2017},
	keywords = {Read},
}

@misc{chung_uncertainty_2021,
	title = {Uncertainty {Toolbox}: an {Open}-{Source} {Library} for {Assessing}, {Visualizing}, and {Improving} {Uncertainty} {Quantification}},
	shorttitle = {Uncertainty {Toolbox}},
	url = {http://arxiv.org/abs/2109.10254},
	abstract = {With increasing deployment of machine learning systems in various real-world tasks, there is a greater need for accurate quantification of predictive uncertainty. While the common goal in uncertainty quantification (UQ) in machine learning is to approximate the true distribution of the target data, many works in UQ tend to be disjoint in the evaluation metrics utilized, and disparate implementations for each metric lead to numerical results that are not directly comparable across different works. To address this, we introduce Uncertainty Toolbox, an open-source python library that helps to assess, visualize, and improve UQ. Uncertainty Toolbox additionally provides pedagogical resources, such as a glossary of key terms and an organized collection of key paper references. We hope that this toolbox is useful for accelerating and uniting research efforts in uncertainty in machine learning.},
	urldate = {2023-10-17},
	publisher = {arXiv},
	author = {Chung, Youngseog and Char, Ian and Guo, Han and Schneider, Jeff and Neiswanger, Willie},
	month = sep,
	year = {2021},
	note = {arXiv:2109.10254 [cs, stat]},
	keywords = {Computer Science - Machine Learning, Read, Statistics - Machine Learning},
}

@book{soize_uncertainty_2017,
	title = {Uncertainty {Quantification}},
	volume = {47},
	isbn = {978-3-319-54338-3},
	abstract = {This book presents the fundamental notions and advanced mathematical tools in the stochastic modeling of uncertainties and their quantification for large-scale computational models in sciences and engineering. In particular, it focuses in parametric uncertainties, and non-parametric uncertainties with applications from the structural dynamics and vibroacoustics of complex mechanical systems, from micromechanics and multiscale mechanics of heterogeneous materials. 

Resulting from a course developed by the author, the book begins with a description of the fundamental mathematical tools of probability and statistics that are directly useful for uncertainty quantification. It proceeds with a well carried out description of some basic and advanced methods for constructing stochastic models of uncertainties, paying particular attention to the problem of calibrating and identifying a stochastic model of uncertainty when experimental data is available. {\textless}
This book is intended to be a graduate-level textbook for students as well as professionals interested in the theory, computation, and applications of risk and prediction in science and engineering fields.},
	author = {Soize, Christian},
	month = jan,
	year = {2017},
	doi = {10.1007/978-3-319-54339-0},
	keywords = {Read},
}

@inproceedings{ashukha_pitfalls_2020,
	title = {Pitfalls of {In}-{Domain} {Uncertainty} {Estimation} and {Ensembling} in {Deep} {Learning}},
	url = {https://iclr.cc/virtual_2020/poster_BJxI5gHKDr.html},
	abstract = {Uncertainty estimation and ensembling methods go hand-in-hand. Uncertainty estimation is one of the main benchmarks for assessment of ensembling performance. At the same time, deep learning ensembles have provided state-of-the-art results in uncertainty estimation. In this work, we focus on in-domain uncertainty for image classification. We explore the standards for its quantification and point out pitfalls of existing metrics. Avoiding these pitfalls, we perform a broad study of different ensembling techniques. To provide more insight in this study, we introduce the deep ensemble equivalent score (DEE) and show that many sophisticated ensembling techniques are equivalent to an ensemble of only few independently trained networks in terms of test performance.},
	language = {en},
	urldate = {2023-10-17},
	author = {Ashukha, Arsenii and Lyzhov, Alexander and Molchanov, Dmitry and Vetrov, Dmitry},
	month = apr,
	year = {2020},
	keywords = {unread},
}

@inproceedings{klas_uncertainty_2018,
	address = {Cham},
	series = {Lecture {Notes} in {Computer} {Science}},
	title = {Uncertainty in {Machine} {Learning} {Applications}: {A} {Practice}-{Driven} {Classification} of {Uncertainty}},
	isbn = {978-3-319-99229-7},
	shorttitle = {Uncertainty in {Machine} {Learning} {Applications}},
	doi = {10.1007/978-3-319-99229-7_36},
	abstract = {Software-intensive systems that rely on machine learning (ML) and artificial intelligence (AI) are increasingly becoming part of our daily life, e.g., in recommendation systems or semi-autonomous vehicles. However, the use of ML and AI is accompanied by uncertainties regarding their outcomes. Dealing with such uncertainties is particularly important when the actions of these systems can harm humans or the environment, such as in the case of a medical product or self-driving car. To enable a system to make informed decisions when confronted with the uncertainty of embedded AI/ML models and possible safety-related consequences, these models do not only have to provide a defined functionality but must also describe as precisely as possible the likelihood of their outcome being wrong or outside a given range of accuracy. Thus, this paper proposes a classification of major uncertainty sources that is usable and useful in practice: scope compliance, data quality, and model fit. In particular, we highlight the implications of these classes in the development and testing of ML and AI models by establishing links to specific activities during development and testing and means for quantifying and dealing with these different sources of uncertainty.},
	language = {en},
	booktitle = {Computer {Safety}, {Reliability}, and {Security}},
	publisher = {Springer International Publishing},
	author = {Kläs, Michael and Vollmer, Anna Maria},
	editor = {Gallina, Barbara and Skavhaug, Amund and Schoitsch, Erwin and Bitsch, Friedemann},
	year = {2018},
	keywords = {Artificial intelligence, Data quality, Dependability, Empirical modelling, Model validation, Safety engineering, unread},
	pages = {431--438},
}

@misc{noauthor_uncertainty_nodate,
	title = {Uncertainty {Quantification}: {An} {Accelerated} {Course} with {Advanced} {Applications} in {Computational} {Engineering} {\textbar} {SpringerLink}},
	url = {https://link.springer.com/book/10.1007/978-3-319-54339-0},
	urldate = {2023-10-11},
	keywords = {unread},
}

@misc{zeinhofer_unified_2023,
	title = {A {Unified} {Framework} for the {Error} {Analysis} of {Physics}-{Informed} {Neural} {Networks}},
	url = {http://arxiv.org/abs/2311.00529},
	abstract = {We prove a priori and a posteriori error estimates, also known as the generalization error in the machine learning community, for physics-informed neural networks (PINNs) for linear PDEs. We analyze elliptic equations in primal and mixed form, elasticity, parabolic, hyperbolic and Stokes equations; and a PDE constrained optimization problem. For the analysis, we propose an abstract framework in the common language of bilinear forms, and we show that coercivity and continuity lead to error estimates. Our results give insight into the potential of neural networks for high dimensional PDEs and into the benefit of encoding constraints directly in the ansatz class. The provided estimates are -- apart from the Poisson equation -- the first results of best-approximation and a posteriori error-control type. Finally, utilizing recent advances in PINN optimization, we present numerical examples that illustrate the ability of the method to achieve accurate solutions.},
	urldate = {2023-11-04},
	publisher = {arXiv},
	author = {Zeinhofer, Marius and Masri, Rami and Mardal, Kent-André},
	month = nov,
	year = {2023},
	note = {arXiv:2311.00529 [cs, math]},
	keywords = {65M12, 65M15, Mathematics - Numerical Analysis, unread},
}

@article{hullermeier_aleatoric_2021,
	title = {Aleatoric and {Epistemic} {Uncertainty} in {Machine} {Learning}: {An} {Introduction} to {Concepts} and {Methods}},
	volume = {110},
	issn = {0885-6125, 1573-0565},
	shorttitle = {Aleatoric and {Epistemic} {Uncertainty} in {Machine} {Learning}},
	url = {http://arxiv.org/abs/1910.09457},
	doi = {10.1007/s10994-021-05946-3},
	abstract = {The notion of uncertainty is of major importance in machine learning and constitutes a key element of machine learning methodology. In line with the statistical tradition, uncertainty has long been perceived as almost synonymous with standard probability and probabilistic predictions. Yet, due to the steadily increasing relevance of machine learning for practical applications and related issues such as safety requirements, new problems and challenges have recently been identified by machine learning scholars, and these problems may call for new methodological developments. In particular, this includes the importance of distinguishing between (at least) two different types of uncertainty, often referred to as aleatoric and epistemic. In this paper, we provide an introduction to the topic of uncertainty in machine learning as well as an overview of attempts so far at handling uncertainty in general and formalizing this distinction in particular.},
	number = {3},
	urldate = {2023-10-17},
	journal = {Machine Learning},
	author = {Hüllermeier, Eyke and Waegeman, Willem},
	month = mar,
	year = {2021},
	note = {arXiv:1910.09457 [cs, stat]},
	keywords = {Computer Science - Machine Learning, Statistics - Machine Learning, unread},
	pages = {457--506},
}

@misc{schweighofer_introducing_2023-1,
	title = {Introducing an {Improved} {Information}-{Theoretic} {Measure} of {Predictive} {Uncertainty}},
	url = {http://arxiv.org/abs/2311.08309},
	abstract = {Applying a machine learning model for decision-making in the real world requires to distinguish what the model knows from what it does not. A critical factor in assessing the knowledge of a model is to quantify its predictive uncertainty. Predictive uncertainty is commonly measured by the entropy of the Bayesian model average (BMA) predictive distribution. Yet, the properness of this current measure of predictive uncertainty was recently questioned. We provide new insights regarding those limitations. Our analyses show that the current measure erroneously assumes that the BMA predictive distribution is equivalent to the predictive distribution of the true model that generated the dataset. Consequently, we introduce a theoretically grounded measure to overcome these limitations. We experimentally verify the benefits of our introduced measure of predictive uncertainty. We find that our introduced measure behaves more reasonably in controlled synthetic tasks. Moreover, our evaluations on ImageNet demonstrate that our introduced measure is advantageous in real-world applications utilizing predictive uncertainty.},
	urldate = {2023-11-21},
	publisher = {arXiv},
	author = {Schweighofer, Kajetan and Aichberger, Lukas and Ielanskyi, Mykyta and Hochreiter, Sepp},
	month = nov,
	year = {2023},
	note = {arXiv:2311.08309 [cs, stat]},
	keywords = {Computer Science - Machine Learning, Statistics - Machine Learning},
}

@article{ashukha_pitfalls_2020-1,
	title = {{PITFALLS} {OF} {IN}-{DOMAIN} {UNCERTAINTY} {ESTIMATION} {AND} {ENSEMBLING} {IN} {DEEP} {LEARNING}},
	abstract = {Uncertainty estimation and ensembling methods go hand-in-hand. Uncertainty estimation is one of the main benchmarks for assessment of ensembling performance. At the same time, deep learning ensembles have provided state-of-the-art results in uncertainty estimation. In this work, we focus on in-domain uncertainty for image classiﬁcation. We explore the standards for its quantiﬁcation and point out pitfalls of existing metrics. Avoiding these pitfalls, we perform a broad study of different ensembling techniques. To provide more insight in this study, we introduce the deep ensemble equivalent score (DEE) and show that many sophisticated ensembling techniques are equivalent to an ensemble of only few independently trained networks in terms of test performance.},
	language = {en},
	author = {Ashukha, Arsenii and Molchanov, Dmitry and Lyzhov, Alexander and Vetrov, Dmitry},
	year = {2020},
}

@inproceedings{ovadia_can_2019,
	title = {Can you trust your model' s uncertainty? {Evaluating} predictive uncertainty under dataset shift},
	volume = {32},
	shorttitle = {Can you trust your model' s uncertainty?},
	url = {https://proceedings.neurips.cc/paper_files/paper/2019/hash/8558cb408c1d76621371888657d2eb1d-Abstract.html},
	abstract = {Modern machine learning methods including deep learning have achieved great success in predictive accuracy for supervised learning tasks, but may still fall short in giving useful estimates of their predictive uncertainty. Quantifying uncertainty is especially critical in real-world settings, which often involve input distributions that are shifted from the training distribution due to a variety of factors including sample bias and non-stationarity.  In such settings, well calibrated uncertainty estimates convey information about when a model's output should (or should not) be trusted.  Many probabilistic deep learning methods, including Bayesian-and non-Bayesian methods, have been proposed in the literature for quantifying predictive uncertainty, but to our knowledge there has not previously been a rigorous large-scale empirical comparison of these methods under dataset shift. We present a large-scale benchmark of existing state-of-the-art methods on classification problems and investigate the effect of dataset shift on accuracy and calibration.  We find that traditional post-hoc calibration does indeed fall short, as do several other previous methods.  However, some methods that marginalize over models give surprisingly strong results across a broad spectrum of tasks.},
	urldate = {2023-10-17},
	booktitle = {Advances in {Neural} {Information} {Processing} {Systems}},
	publisher = {Curran Associates, Inc.},
	author = {Ovadia, Yaniv and Fertig, Emily and Ren, Jie and Nado, Zachary and Sculley, D. and Nowozin, Sebastian and Dillon, Joshua and Lakshminarayanan, Balaji and Snoek, Jasper},
	year = {2019},
}
