@unpublished{alvarezKernelsVectorValuedFunctions2012,
  title = {Kernels for {{Vector-Valued Functions}}: A {{Review}}},
  shorttitle = {Kernels for {{Vector-Valued Functions}}},
  author = {Alvarez, Mauricio A. and Rosasco, Lorenzo and Lawrence, Neil D.},
  date = {2012-04-16},
  eprint = {1106.6251},
  eprinttype = {arxiv},
  eprintclass = {cs, math, stat},
  url = {http://arxiv.org/abs/1106.6251},
  urldate = {2020-02-11},
  abstract = {Kernel methods are among the most popular techniques in machine learning. From a frequentist/discriminative perspective they play a central role in regularization theory as they provide a natural choice for the hypotheses space and the regularization functional through the notion of reproducing kernel Hilbert spaces. From a Bayesian/generative perspective they are the key in the context of Gaussian processes, where the kernel function is also known as the covariance function. Traditionally, kernel methods have been used in supervised learning problem with scalar outputs and indeed there has been a considerable amount of work devoted to designing and learning kernels. More recently there has been an increasing interest in methods that deal with multiple outputs, motivated partly by frameworks like multitask learning. In this paper, we review different methods to design or learn valid kernel functions for multiple outputs, paying particular attention to the connection between probabilistic and functional methods.},
  keywords = {Computer Science - Artificial Intelligence,Mathematics - Statistics Theory,Statistics - Machine Learning},
  file = {/home/steffen/OneDrive/Library/PhD/Kernels/Alvarez_et_al-2012-Kernels_for_Vector-Valued_Functions.pdf;/home/steffen/Zotero/storage/A6CWCKSB/1106.html}
}

@misc{banksUncertaintyPropagationQuantification2012,
  title = {Uncertainty {{Propagation}} and {{Quantification}} in a {{Continuous Time Dynamical System}}},
  author = {Banks, H. T. and Hu, Shuhua},
  date = {2012},
  abstract = {Uncertainty propagation and quantification has gained considerable research at-tention during recent years. In this paper we consider uncertainty propagation and quantification in a continuous-time dynamical system governed by ordinary differential equations with uncertain/stochastic components. Specifically, we focus on the time evolution of probability density functions of the resulting stochastic processes, and dis-cuss their applications in different fields with particular focus on population dynamics. In addition, we compare the difference in the stochastic processes resulting from dif-ferential equations with different types of random inputs, and discuss the connections among them. Key Words: stochastic differential equations, random differential equations, probability density functions, Fokker-Planck equation, Liouville equation, structured population model, growth rate distributed structured population model, Markov operators and semigroups, pointwise equivalence. 1},
  file = {/home/steffen/OneDrive/Library/PhD/Prob-Solver/Random Dynamics/Banks_Hu-2012-Uncertainty_Propagation_and_Quantification_in_a.pdf;/home/steffen/Zotero/storage/KKW7IHDN/summary.html}
}

@article{bijlOnlineSparseGaussian2015,
  title = {Online Sparse {{Gaussian}} Process Regression Using {{FITC}} and {{PITC}} Approximations},
  shorttitle = {Online Sparse {{Gaussian}} Process Regression Using {{FITC}} and {{PITC}} Approximations},
  author = {Bijl, Hildo and van Wingerden, Jan-Willem and B. Schön, Thomas and Verhaegen, Michel},
  options = {useprefix=true},
  date = {2015-01-01},
  journaltitle = {IFAC-PapersOnLine},
  shortjournal = {IFAC-PapersOnLine},
  series = {17th {{IFAC Symposium}} on {{System Identification SYSID}} 2015},
  volume = {48},
  number = {28},
  pages = {703--708},
  issn = {2405-8963},
  doi = {10.1016/j.ifacol.2015.12.212},
  url = {https://www.sciencedirect.com/science/article/pii/S2405896315028360},
  urldate = {2021-02-16},
  abstract = {We provide a method which allows for online updating of sparse Gaussian Process (GP) regression algorithms for any set of inducing inputs. This method is derived both for the Fully Independent Training Conditional (FITC) and the Partially Independent Training Conditional (PITC) approximation, and it allows the inclusion of a new measurement point xn+1 in O(m2) time, with m denoting the size of the set of inducing inputs. Due to the online nature of the algorithms, it is possible to forget earlier measurement data, which means that also the memory space required is O(m2), both for FITC and PITC. We show that this method is able to efficiently apply GP regression to a large data set with accurate results.},
  langid = {english},
  file = {/home/steffen/OneDrive/Library/PhD/a sort/Bijl_et_al-2015-Online_sparse_Gaussian_process_regression_using.pdf;/home/steffen/Zotero/storage/TC8M4WP7/S2405896315028360.html}
}

@inproceedings{buisson-fenetActivelyLearningGaussian2020,
  title = {Actively Learning Gaussian Process Dynamics},
  booktitle = {Proceedings of the 2nd {{Conference}} on {{Learning}} for {{Dynamics}} and {{Control}}},
  author = {Buisson-Fenet, Mona and Solowjow, Friedrich and Trimpe, Sebastian},
  date = {2020-06-10/2020-06-11},
  series = {Proceedings of Machine Learning Research},
  volume = {120},
  pages = {5--15},
  publisher = {{PMLR}},
  location = {{The Cloud}},
  url = {http://proceedings.mlr.press/v120/buisson-fenet20a.html},
  abstract = {Despite the availability of ever more data enabled through modern sensor and computer technology, it still remains an open problem to learn dynamical systems in a sample-efficient way.We propose active learning strategies that leverage information-theoretical properties arising naturally during Gaussian process regression, while respecting constraints on the sampling process imposed by the system dynamics. Sample points are selected in regions with high uncertainty, leading to exploratory behavior and data-efficient training of the model.All results are verified in an extensive numerical benchmark.},
  pdf = {http://proceedings.mlr.press/v120/buisson-fenet20a/buisson-fenet20a.pdf},
  file = {/home/steffen/OneDrive/Library/PhD/Motivation/Active Learning/Buisson-Fenet_et_al-2020-Actively_learning_gaussian_process_dynamics.pdf}
}

@article{davisRowModificationsSparse2005,
  title = {Row {{Modifications}} of a {{Sparse Cholesky Factorization}}},
  author = {Davis, Timothy A. and Hager, William W.},
  date = {2005-01},
  journaltitle = {SIAM Journal on Matrix Analysis and Applications},
  shortjournal = {SIAM J. Matrix Anal. \& Appl.},
  volume = {26},
  number = {3},
  pages = {621--639},
  issn = {0895-4798, 1095-7162},
  doi = {10.1137/S089547980343641X},
  url = {http://epubs.siam.org/doi/10.1137/S089547980343641X},
  urldate = {2021-02-22},
  abstract = {Given a sparse, symmetric positive definite matrix C and an associated sparse Cholesky factorization LDLT, we develop sparse techniques for updating the factorization after a symmetric modification of a row and column of C. We show how the modification in the Cholesky factorization associated with this rank-2 modification of C can be computed efficiently using a sparse rank-1 technique developed in [T. A. Davis and W. W. Hager, SIAM J. Matrix Anal. Appl., 20 (1999), pp. 606–627]. We also determine how the solution of a linear system Lx = b changes after changing a row and column of C or after a rank-r change in C.},
  langid = {english},
  file = {/home/steffen/OneDrive/Library/PhD/a sort/Davis_Hager-2005-Row_Modifications_of_a_Sparse_Cholesky.pdf}
}

@article{delacruzHigherOrderLocal2007,
  title = {A Higher Order Local Linearization Method for Solving Ordinary Differential Equations},
  author = {De la Cruz, H. and Biscay, R.J. and Carbonell, F. and Ozaki, T. and Jimenez, J.C.},
  date = {2007-02},
  journaltitle = {Applied Mathematics and Computation},
  shortjournal = {Applied Mathematics and Computation},
  volume = {185},
  number = {1},
  pages = {197--212},
  issn = {00963003},
  doi = {10.1016/j.amc.2006.06.096},
  url = {https://linkinghub.elsevier.com/retrieve/pii/S0096300306008514},
  urldate = {2022-08-02},
  abstract = {The Local Linearization (LL) method for the integration of ordinary differential equations is an explicit one-step method that has a number of suitable dynamical properties. However, a major drawback of the LL integrator is that its order of convergence is only two. The present paper overcomes this limitation by introducing a new class of numerical integrators, called the LLT method, that is based on the addition of a correction term to the LL approximation. In this way an arbitrary order of convergence can be achieved while retaining the dynamic properties of the LL method. In particular, it is proved that the LLT method reproduces correctly the phase portrait of a dynamical system near hyperbolic stationary points to the order of convergence. The performance of the introduced method is further illustrated through computer simulations.},
  langid = {english},
  file = {/home/steffen/OneDrive/Library/PhD/Prob-Solver/Solvers/De_la_Cruz_et_al-2007-A_higher_order_local_linearization_method_for.pdf}
}

@unpublished{ensingerStructurepreservingGaussianProcess2022,
  title = {Structure-Preserving {{Gaussian Process Dynamics}}},
  author = {Ensinger, Katharina and Solowjow, Friedrich and Ziesche, Sebastian and Tiemann, Michael and Trimpe, Sebastian},
  date = {2022-01-09},
  eprint = {2102.01606},
  eprinttype = {arxiv},
  eprintclass = {cs},
  publisher = {{arXiv}},
  doi = {10.48550/arXiv.2102.01606},
  url = {http://arxiv.org/abs/2102.01606},
  urldate = {2022-05-23},
  abstract = {Most physical processes posses structural properties such as constant energies, volumes, and other invariants over time. When learning models of such dynamical systems, it is critical to respect these invariants to ensure accurate predictions and physically meaningful behavior. Strikingly, state-of-the-art methods in Gaussian process (GP) dynamics model learning are not addressing this issue. On the other hand, classical numerical integrators are specifically designed to preserve these crucial properties through time. We propose to combine the advantages of GPs as function approximators with structure preserving numerical integrators for dynamical systems, such as Runge-Kutta methods. These integrators assume access to the ground truth dynamics and require evaluations of intermediate and future time steps that are unknown in a learning-based scenario. This makes direct inference of the GP dynamics, with embedded numerical scheme, intractable. Our key technical contribution is the evaluation of the implicitly defined Runge-Kutta transition probability. In a nutshell, we introduce an implicit layer for GP regression, which is embedded into a variational inference-based model learning scheme.},
  keywords = {Computer Science - Machine Learning},
  file = {/home/steffen/OneDrive/Library/PhD/Gaussian Processes/GP Propagation/Ensinger_et_al-2022-Structure-preserving_Gaussian_Process_Dynamics.pdf;/home/steffen/Zotero/storage/22AWJ3TG/2102.html}
}

@article{girardGaussianProcessPriors2002,
  title = {Gaussian {{Process Priors}} with {{Uncertain Inputs Application}} to {{Multiple-Step Ahead Time Series Forecasting}}},
  author = {Girard, Agathe and Rasmussen, Carl Edward and Candela, Joaquin Q. and Murray-Smith, Roderick},
  date = {2002},
  journaltitle = {Advances in Neural Information Processing Systems},
  volume = {15},
  url = {https://proceedings.neurips.cc/paper/2002/hash/f3ac63c91272f19ce97c7397825cc15f-Abstract.html},
  urldate = {2021-05-20},
  langid = {english},
  file = {/home/steffen/OneDrive/Library/PhD/Gaussian Processes/GP Propagation/Girard_et_al-2002-Gaussian_Process_Priors_with_Uncertain_Inputs.pdf;/home/steffen/Zotero/storage/BJYSK7FD/f3ac63c91272f19ce97c7397825cc15f-Abstract.html}
}

@article{girardMultiplestepAheadPrediction2003,
  title = {Multiple-Step Ahead Prediction for Non Linear Dynamic Systems – {{A Gaussian Process}} Treatment with Propagation of the Uncertainty},
  author = {Girard, Agathe and Rasmussen, Carl Edward and Murray-Smith, Roderick},
  date = {2003},
  journaltitle = {Advances in Neural Information Processing Systems},
  pages = {9},
  langid = {english},
  file = {/home/steffen/OneDrive/Library/PhD/Gaussian Processes/GP Propagation/Girard_et_al-2003-Multiple-step_ahead_prediction_for_non_linear.pdf}
}

@article{grootMultiplestepTimeSeries2011,
  title = {Multiple-Step {{Time Series Forecasting}} with {{Sparse Gaussian Processes}}},
  author = {Groot, P. and Lucas, P. and Bosch, P.},
  date = {2011},
  journaltitle = {http://allserv.kahosl.be/bnaic2011/},
  publisher = {{[S.l.] : [S.n.]}},
  url = {https://repository.ubn.ru.nl/handle/2066/92100},
  urldate = {2022-08-29},
  abstract = {BNAIC :  the 23rd Benelux Conference on Artificial Intelligence, 3 - 4 November 2011, Gent, Belgium, 03 november 2011},
  langid = {english},
  annotation = {Accepted: 2012-02-19T11:50:24Z},
  file = {/home/steffen/OneDrive/Library/PhD/Gaussian Processes/GP Propagation/Groot_et_al-2011-Multiple-step_Time_Series_Forecasting_with_Sparse.pdf;/home/steffen/Zotero/storage/F6BEA79T/92100.html}
}

@book{guckenheimerNonlinearOscillationsDynamical2013,
  title = {Nonlinear {{Oscillations}}, {{Dynamical Systems}}, and {{Bifurcations}} of {{Vector Fields}}.},
  author = {Guckenheimer, John and Holmes, Philip and Guckenheimer, John and Sirovich, Lawrence},
  date = {2013},
  publisher = {{Springer}},
  location = {{New York, NY}},
  url = {https://public.ebookcentral.proquest.com/choice/publicfullrecord.aspx?p=5575396},
  urldate = {2020-05-18},
  isbn = {978-1-4612-1140-2},
  langid = {english},
  annotation = {OCLC: 1066180459},
  file = {/home/steffen/OneDrive/Library/Library/Dynamics/Guckenheimer et al_2013_Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields.pdf}
}

@inproceedings{hegdeVariationalMultipleShooting2022,
  title = {Variational Multiple Shooting for {{Bayesian ODEs}} with {{Gaussian}} Processes},
  booktitle = {Proceedings of the {{Thirty-Eighth Conference}} on {{Uncertainty}} in {{Artificial Intelligence}}},
  author = {Hegde, Pashupati and Yıldız, Çağatay and Lähdesmäki, Harri and Kaski, Samuel and Heinonen, Markus},
  date = {2022-08-17},
  pages = {790--799},
  publisher = {{PMLR}},
  issn = {2640-3498},
  url = {https://proceedings.mlr.press/v180/hegde22a.html},
  urldate = {2023-05-14},
  abstract = {Recent machine learning advances have proposed black-box estimation of \textbackslash textit\{unknown continuous-time system dynamics\} directly from data. However, earlier works are based on approximative solutions or point estimates. We propose a novel Bayesian nonparametric model that uses Gaussian processes to infer posteriors of unknown ODE systems directly from data. We derive sparse variational inference with decoupled functional sampling to represent vector field posteriors. We also introduce a probabilistic shooting augmentation to enable efficient inference from arbitrarily long trajectories. The method demonstrates the benefit of computing vector field posteriors, with predictive uncertainty scores outperforming alternative methods on multiple ODE learning tasks.},
  eventtitle = {Uncertainty in {{Artificial Intelligence}}},
  langid = {english},
  file = {/home/steffen/OneDrive/Library/PhD/GaussianODE/Hegde_et_al-2022-Variational_multiple_shooting_for_Bayesian_ODEs2.pdf;/home/steffen/OneDrive/Library/PhD/GaussianODE/Hegde_et_al-2022-Variational_multiple_shooting_for_Bayesian_ODEs3.pdf}
}

@unpublished{heinonenLearningUnknownODE2018,
  title = {Learning Unknown {{ODE}} Models with {{Gaussian}} Processes},
  author = {Heinonen, Markus and Yildiz, Cagatay and Mannerström, Henrik and Intosalmi, Jukka and Lähdesmäki, Harri},
  date = {2018-03-12},
  eprint = {1803.04303},
  eprinttype = {arxiv},
  url = {http://arxiv.org/abs/1803.04303},
  urldate = {2019-12-03},
  abstract = {In conventional ODE modelling coefficients of an equation driving the system state forward in time are estimated. However, for many complex systems it is practically impossible to determine the equations or interactions governing the underlying dynamics. In these settings, parametric ODE model cannot be formulated. Here, we overcome this issue by introducing a novel paradigm of nonparametric ODE modelling that can learn the underlying dynamics of arbitrary continuous-time systems without prior knowledge. We propose to learn non-linear, unknown differential functions from state observations using Gaussian process vector fields within the exact ODE formalism. We demonstrate the model's capabilities to infer dynamics from sparse data and to simulate the system forward into future.},
  langid = {english},
  keywords = {\_tablet,Important,ODE,Statistics - Machine Learning},
  file = {/home/steffen/OneDrive/Library/PhD/GaussianMPC/Heinonen_et_al-2018-Learning_unknown_ODE_models_with_Gaussian.pdf}
}

@book{hennigProbabilisticNumericsComputation2022,
  title = {Probabilistic {{Numerics}}: {{Computation}} as {{Machine Learning}}},
  shorttitle = {Probabilistic {{Numerics}}},
  author = {Hennig, Philipp and Osborne, Michael A. and Kersting, Hans P.},
  date = {2022},
  publisher = {{Cambridge University Press}},
  location = {{Cambridge}},
  doi = {10.1017/9781316681411},
  url = {https://www.cambridge.org/core/books/probabilistic-numerics/0EBFF0B15E2481099F6EED1F62EE1ABE},
  urldate = {2023-02-14},
  abstract = {Probabilistic numerical computation formalises the connection between machine learning and applied mathematics. Numerical algorithms approximate intractable quantities from computable ones. They estimate integrals from evaluations of the integrand, or the path of a dynamical system described by differential equations from evaluations of the vector field. In other words, they infer a latent quantity from data. This book shows that it is thus formally possible to think of computational routines as learning machines, and to use the notion of Bayesian inference to build more flexible, efficient, or customised algorithms for computation. The text caters for Masters' and PhD students, as well as postgraduate researchers in artificial intelligence, computer science, statistics, and applied mathematics. Extensive background material is provided along with a wealth of figures, worked examples, and exercises (with solutions) to develop intuition.},
  isbn = {978-1-107-16344-7},
  file = {/home/steffen/OneDrive/Library/PhD/Prob-Solver/Probablistics/Hennig et al_2022_Probabilistic Numerics.pdf;/home/steffen/Zotero/storage/D5PZBBUS/0EBFF0B15E2481099F6EED1F62EE1ABE.html}
}

@inproceedings{hewingSimulationTrajectoryPrediction2020,
  title = {On Simulation and Trajectory Prediction with Gaussian Process Dynamics},
  booktitle = {Proceedings of the 2nd {{Conference}} on {{Learning}} for {{Dynamics}} and {{Control}}},
  author = {Hewing, Lukas and Arcari, Elena and Fröhlich, Lukas P. and Zeilinger, Melanie N.},
  date = {2020-06-10/2020-06-11},
  series = {Proceedings of Machine Learning Research},
  volume = {120},
  pages = {424--434},
  publisher = {{PMLR}},
  location = {{The Cloud}},
  url = {http://proceedings.mlr.press/v120/hewing20a.html},
  abstract = {Established techniques for simulation and prediction with Gaussian process (GP) dynamics implicitly make use of an independence assumption on successive function evaluations of the dynamics model. This can result in significant error and underestimation of the prediction uncertainty, potentially leading to failures in safety-critical applications. This paper proposes methods that explicitly take the correlation of successive function evaluations into account. We first describe two sampling-based techniques; one approach provides samples of the true trajectory distribution, suitable for ‘ground truth’ simulations, while the other draws function samples from basis function approximations of the GP. Second, we present a linearization-based technique that directly provides approximations of the trajectory distribution, taking correlations explicitly into account. We demonstrate the procedures in simple numerical examples, contrasting the results with established methods.},
  pdf = {http://proceedings.mlr.press/v120/hewing20a/hewing20a.pdf},
  file = {/home/steffen/OneDrive/Library/PhD/a sort/Hewing_et_al-2020-On_simulation_and_trajectory_prediction_with.pdf}
}

@article{hochbruckExponentialIntegrators2010,
  title = {Exponential Integrators},
  author = {Hochbruck, Marlis and Ostermann, Alexander},
  date = {2010-05},
  journaltitle = {Acta Numerica},
  shortjournal = {Acta Numerica},
  volume = {19},
  pages = {209--286},
  issn = {0962-4929, 1474-0508},
  doi = {10.1017/S0962492910000048},
  url = {https://www.cambridge.org/core/product/identifier/S0962492910000048/type/journal_article},
  urldate = {2022-08-03},
  abstract = {In this paper we consider the construction, analysis, implementation and application of exponential integrators. The focus will be on two types of stiff problems. The first one is characterized by a Jacobian that possesses eigenvalues with large negative real parts. Parabolic partial differential equations and their spatial discretization are typical examples. The second class consists of highly oscillatory problems with purely imaginary eigenvalues of large modulus. Apart from motivating the construction of exponential integrators for various classes of problems, our main intention in this article is to present the mathematics behind these methods. We will derive error bounds that are independent of stiffness or highest frequencies in the system.             Since the implementation of exponential integrators requires the evaluation of the product of a matrix function with a vector, we will briefly discuss some possible approaches as well. The paper concludes with some applications, in which exponential integrators are used.},
  langid = {english},
  file = {/home/steffen/OneDrive/Library/PhD/Prob-Solver/Solvers/Hochbruck_Ostermann-2010-Exponential_integrators.pdf}
}

@unpublished{ialongoOvercomingMeanFieldApproximations2019,
  title = {Overcoming {{Mean-Field Approximations}} in {{Recurrent Gaussian Process Models}}},
  author = {Ialongo, Alessandro Davide and van der Wilk, Mark and Hensman, James and Rasmussen, Carl Edward},
  options = {useprefix=true},
  date = {2019-06-13},
  eprint = {1906.05828},
  eprinttype = {arxiv},
  eprintclass = {cs, stat},
  url = {http://arxiv.org/abs/1906.05828},
  urldate = {2020-06-19},
  abstract = {We identify a new variational inference scheme for dynamical systems whose transition function is modelled by a Gaussian process. Inference in this setting has either employed computationally intensive MCMC methods, or relied on factorisations of the variational posterior. As we demonstrate in our experiments, the factorisation between latent system states and transition function can lead to a miscalibrated posterior and to learning unnecessarily large noise terms. We eliminate this factorisation by explicitly modelling the dependence between state trajectories and the Gaussian process posterior. Samples of the latent states can then be tractably generated by conditioning on this representation. The method we obtain (VCDT: variationally coupled dynamics and trajectories) gives better predictive performance and more calibrated estimates of the transition function, yet maintains the same time and space complexities as mean-field methods. Code is available at: github.com/ialong/GPt.},
  langid = {english},
  keywords = {Computer Science - Machine Learning,Statistics - Machine Learning},
  file = {/home/steffen/OneDrive/Library/PhD/GaussianMPC/Ialongo_et_al-2019-Overcoming_Mean-Field_Approximations_in_Recurrent.pdf}
}

@unpublished{kamtheDataEfficientReinforcementLearning2017,
  title = {Data-{{Efficient Reinforcement Learning}} with {{Probabilistic Model Predictive Control}}},
  author = {Kamthe, Sanket and Deisenroth, Marc Peter},
  date = {2017-06-20},
  eprint = {1706.06491},
  eprinttype = {arxiv},
  url = {http://arxiv.org/abs/1706.06491},
  urldate = {2019-10-17},
  abstract = {Trial-and-error based reinforcement learning (RL) has seen rapid advancements in recent times, especially with the advent of deep neural networks. However, the majority of autonomous RL algorithms require a large number of interactions with the environment. A large number of interactions may be impractical in many real-world applications, such as robotics, and many practical systems have to obey limitations in the form of state space or control constraints. To reduce the number of system interactions while simultaneously handling constraints, we propose a model-based RL framework based on probabilistic Model Predictive Control (MPC). In particular, we propose to learn a probabilistic transition model using Gaussian Processes (GPs) to incorporate model uncertainty into long-term predictions, thereby, reducing the impact of model errors. We then use MPC to find a control sequence that minimises the expected long-term cost. We provide theoretical guarantees for first-order optimality in the GP-based transition models with deterministic approximate inference for long-term planning. We demonstrate that our approach does not only achieve state-of-the-art data efficiency, but also is a principled way for RL in constrained environments.},
  keywords = {Electrical Engineering and Systems Science - Systems and Control,Statistics - Machine Learning},
  file = {/home/steffen/OneDrive/Library/PhD/Motivation/State Space GPs/Kamthe_Deisenroth-2017-Data-Efficient_Reinforcement_Learning_with.pdf;/home/steffen/Zotero/storage/QSX3B4WU/1706.html}
}

@inproceedings{kerstingActiveUncertaintyCalibration2016,
  title = {Active Uncertainty Calibration in {{Bayesian ODE}} Solvers},
  booktitle = {Proceedings of the {{Thirty-Second Conference}} on {{Uncertainty}} in {{Artificial Intelligence}}},
  author = {Kersting, Hans and Hennig, Philipp},
  date = {2016-06-25},
  series = {{{UAI}}'16},
  pages = {309--318},
  publisher = {{AUAI Press}},
  location = {{Arlington, Virginia, USA}},
  abstract = {There is resurging interest, in statistics and machine learning, in solvers for ordinary differential equations (ODEs) that return probability measures instead of point estimates. Recently, Conrad et al. introduced a sampling-based class of methods that are 'well-calibrated' in a specific sense. But the computational cost of these methods is significantly above that of classic methods. On the other hand, Schober et al. pointed out a precise connection between classic Runge-Kutta ODE solvers and Gaussian filters, which gives only a rough probabilistic calibration, but at negligible cost overhead. By formulating the solution of ODEs as approximate inference in linear Gaussian SDEs, we investigate a range of probabilistic ODE solvers, that bridge the tradeoff between computational cost and probabilistic calibration, and identify the inaccurate gradient measurement as the crucial source of uncertainty. We propose the novel filtering-based method Bayesian Quadrature filtering (BQF) which uses Bayesian quadrature to actively learn the imprecision in the gradient measurement by collecting multiple gradient evaluations.},
  isbn = {978-0-9966431-1-5},
  file = {/home/steffen/OneDrive/Library/PhD/ProbablisticSolvers/Kersting_Hennig-2016-Active_uncertainty_calibration_in_Bayesian_ODE.pdf}
}

@book{kocijanModellingControlDynamic2015,
  title = {Modelling and Control of Dynamic Systems Using Gaussian Process Models},
  author = {Kocijan, J.},
  date = {2015},
  series = {Advances in Industrial Control},
  publisher = {{Springer International Publishing}},
  url = {https://books.google.de/books?id=_Gn_CgAAQBAJ},
  isbn = {978-3-319-21021-6},
  lccn = {2015954342},
  file = {/home/steffen/OneDrive/Library/PhD/GaussianMPC/Kocijan-2015-Modelling_and_control_of_dynamic_systems_using.pdf}
}

@unpublished{nghiemLinearizedGaussianProcesses2019,
  title = {Linearized {{Gaussian Processes}} for {{Fast Data-driven Model Predictive Control}}},
  author = {Nghiem, Truong X.},
  date = {2019-10-01},
  eprint = {1812.10579},
  eprinttype = {arxiv},
  eprintclass = {cs},
  url = {http://arxiv.org/abs/1812.10579},
  urldate = {2020-08-14},
  abstract = {Data-driven Model Predictive Control (MPC), where the system model is learned from data with machine learning, has recently gained increasing interests in the control community. Gaussian Processes (GP), as a type of statistical models, are particularly attractive due to their modeling flexibility and their ability to provide probabilistic estimates of prediction uncertainty. GP-based MPC has been developed and applied, however the optimization problem is typically non-convex and highly demanding, and scales poorly with model size. This causes unsatisfactory solving performance, even with state-of-the-art solvers, and makes the approach less suitable for real-time control. We develop a method based on a new concept, called linearized Gaussian Process, and Sequential Convex Programming, that can significantly improve the solving performance of GP-based MPC. Our method is not only faster but also much more scalable and predictable than other commonly used methods, as it is much less influenced by the model size. The efficiency and advantages of the algorithm are demonstrated clearly in a numerical example.},
  langid = {english},
  keywords = {Electrical Engineering and Systems Science - Systems and Control},
  file = {/home/steffen/OneDrive/Library/PhD/GaussianMPC/Nghiem-2019-Linearized_Gaussian_Processes_for_Fast.pdf}
}

@thesis{ottoTopologyVectorFields1983,
  title = {Topology of {{Vector Fields}} with {{Uncertainty}}},
  author = {Otto, Mathias},
  date = {1983-01-02},
  institution = {{Otto-von-Guericke-Universität Magdeburg}},
  url = {https://core.ac.uk/download/pdf/51449137.pdf},
  urldate = {2021-02-02},
  file = {/home/steffen/OneDrive/Library/PhD/a sort/Otto-1983-Topology_of_Vector_Fields_with_Uncertainty.pdf}
}

@unpublished{polymenakosSafetyGuaranteesPlanning2020,
  title = {Safety {{Guarantees}} for {{Planning Based}} on {{Iterative Gaussian Processes}}},
  author = {Polymenakos, Kyriakos and Laurenti, Luca and Patane, Andrea and Calliess, Jan-Peter and Cardelli, Luca and Kwiatkowska, Marta and Abate, Alessandro and Roberts, Stephen},
  date = {2020-09-07},
  eprint = {1912.00071},
  eprinttype = {arxiv},
  eprintclass = {cs, stat},
  url = {http://arxiv.org/abs/1912.00071},
  urldate = {2021-03-14},
  abstract = {Gaussian Processes (GPs) are widely employed in control and learning because of their principled treatment of uncertainty. However, tracking uncertainty for iterative, multi-step predictions in general leads to an analytically intractable problem. While approximation methods exist, they do not come with guarantees, making it difficult to estimate their reliability and to trust their predictions. In this work, we derive formal probability error bounds for iterative prediction and planning with GPs. Building on GP properties, we bound the probability that random trajectories lie in specific regions around the predicted values. Namely, given a tolerance \$\textbackslash epsilon {$>$} 0 \$, we compute regions around the predicted trajectory values, such that GP trajectories are guaranteed to lie inside them with probability at least \$1-\textbackslash epsilon\$. We verify experimentally that our method tracks the predictive uncertainty correctly, even when current approximation techniques fail. Furthermore, we show how the proposed bounds can be employed within a safe reinforcement learning framework to verify the safety of candidate control policies, guiding the synthesis of provably safe controllers.},
  langid = {english},
  keywords = {Computer Science - Machine Learning,Statistics - Machine Learning},
  file = {/home/steffen/OneDrive/Library/PhD/a sort/Polymenakos_et_al-2020-Safety_Guarantees_for_Planning_Based_on_Iterative.pdf}
}

@unpublished{rackauckasUniversalDifferentialEquations2020,
  title = {Universal {{Differential Equations}} for {{Scientific Machine Learning}}},
  author = {Rackauckas, Christopher and Ma, Yingbo and Martensen, Julius and Warner, Collin and Zubov, Kirill and Supekar, Rohit and Skinner, Dominic and Ramadhan, Ali},
  date = {2020-01-13},
  eprint = {2001.04385},
  eprinttype = {arxiv},
  url = {http://arxiv.org/abs/2001.04385},
  urldate = {2020-03-11},
  abstract = {In the context of science, the well-known adage "a picture is worth a thousand words" might well be "a model is worth a thousand datasets." Scientific models, such as Newtonian physics or biological gene regulatory networks, are human-driven simplifications of complex phenomena that serve as surrogates for the countless experiments that validated the models. Recently, machine learning has been able to overcome the inaccuracies of approximate modeling by directly learning the entire set of nonlinear interactions from data. However, without any predetermined structure from the scientific basis behind the problem, machine learning approaches are flexible but data-expensive, requiring large databases of homogeneous labeled training data. A central challenge is reconciling data that is at odds with simplified models without requiring "big data". In this work we develop a new methodology, universal differential equations (UDEs), which augments scientific models with machine-learnable structures for scientifically-based learning. We show how UDEs can be utilized to discover previously unknown governing equations, accurately extrapolate beyond the original data, and accelerate model simulation, all in a time and data-efficient manner. This advance is coupled with open-source software that allows for training UDEs which incorporate physical constraints, delayed interactions, implicitly-defined events, and intrinsic stochasticity in the model. Our examples show how a diverse set of computationally-difficult modeling issues across scientific disciplines, from automatically discovering biological mechanisms to accelerating climate simulations by 15,000x, can be handled by training UDEs.},
  keywords = {Computer Science - Machine Learning,Mathematics - Dynamical Systems,Quantitative Biology - Quantitative Methods,Statistics - Machine Learning},
  file = {/home/steffen/OneDrive/Library/PhD/Motivation/Rackauckas_et_al-2020-Universal_Differential_Equations_for_Scientific.pdf;/home/steffen/Zotero/storage/5UQR8I3G/2001.html}
}

@inproceedings{rahimiRandomFeaturesLargeScale2007,
  title = {Random {{Features}} for {{Large-Scale Kernel Machines}}},
  booktitle = {Advances in {{Neural Information Processing Systems}}},
  author = {Rahimi, Ali and Recht, Benjamin},
  date = {2007},
  volume = {20},
  publisher = {{Curran Associates, Inc.}},
  url = {https://papers.nips.cc/paper/2007/hash/013a006f03dbc5392effeb8f18fda755-Abstract.html},
  urldate = {2022-06-08},
  abstract = {To accelerate the training of kernel machines, we propose to map the input data to a randomized low-dimensional feature space and then apply existing fast linear methods. The features are designed so that the inner products of the transformed data are approximately equal to those in the feature space of a user specified shift- invariant kernel. We explore two sets of random features, provide convergence bounds on their ability to approximate various radial basis kernels, and show that in large-scale classification and regression tasks linear machine learning al- gorithms applied to these features outperform state-of-the-art large-scale kernel machines.},
  file = {/home/steffen/OneDrive/Library/PhD/ProbablisticSolvers/Sampling/Rahimi_Recht-2007-Random_Features_for_Large-Scale_Kernel_Machines.pdf}
}

@book{rasmussenGaussianProcessesMachine2006,
  title = {Gaussian Processes for Machine Learning},
  author = {Rasmussen, Carl Edward and Williams, Christopher K. I.},
  date = {2006},
  series = {Adaptive Computation and Machine Learning},
  publisher = {{MIT Press}},
  location = {{Cambridge, Mass}},
  isbn = {978-0-262-18253-9},
  langid = {english},
  pagetotal = {248},
  keywords = {Data processing,Gaussian processes,Machine learning,Mathematical models},
  annotation = {OCLC: ocm61285753},
  file = {/home/steffen/OneDrive/Library/PhD/Gaussian Processes/Rasmussen_Williams-2006-Gaussian_processes_for_machine_learning.pdf}
}

@inproceedings{ridderbuschLearningODEModels2021,
  title = {Learning {{ODE Models}} with {{Qualitative Structure Using Gaussian Processes}}},
  author = {Ridderbusch, Steffen and Offen, Christian and Ober-Blöbaum, Sina and Goulart, Paul},
  date = {2021-12},
  pages = {2896--2896},
  issn = {2576-2370},
  doi = {10.1109/CDC45484.2021.9683426},
  abstract = {Recently there has been increasing interest in the use of learning techniques to model dynamical systems directly from data for scientific and engineering applications. However, in many contexts explicit data collection is expensive and learning algorithms must be data-efficient to be practicable. This suggests using additional qualitative information about the system, which is often available from prior experiments or domain-specific knowledge. We propose an approach to learning a vector field of differential equations using sparse Gaussian Processes that allows us to combine data and additional structural information, like Lie Group symmetries and fixed points. We show that this combination improves extrapolation and long-term behaviour, and reduces computational cost.},
  eventtitle = {2021 60th {{IEEE Conference}} on {{Decision}} and {{Control}} ({{CDC}})},
  keywords = {Computational modeling,Conferences,Data collection,Differential equations,Extrapolation,Gaussian processes,Heuristic algorithms},
  file = {/home/steffen/OneDrive/Library/my papers/published/Ridderbusch et al_2021_Learning ODE Models with Qualitative Structure Using Gaussian Processes.pdf}
}

@article{schoberProbabilisticModelNumerical2019,
  title = {A Probabilistic Model for the Numerical Solution of Initial Value Problems},
  author = {Schober, Michael and Särkkä, Simo and Hennig, Philipp},
  date = {2019-01-01},
  journaltitle = {Statistics and Computing},
  shortjournal = {Stat Comput},
  volume = {29},
  number = {1},
  pages = {99--122},
  issn = {1573-1375},
  doi = {10.1007/s11222-017-9798-7},
  url = {https://doi.org/10.1007/s11222-017-9798-7},
  urldate = {2020-06-01},
  abstract = {We study connections between ordinary differential equation (ODE) solvers and probabilistic regression methods in statistics. We provide a new view of probabilistic ODE solvers as active inference agents operating on stochastic differential equation models that estimate the unknown initial value problem (IVP) solution from approximate observations of the solution derivative, as provided by the ODE dynamics. Adding to this picture, we show that several multistep methods of Nordsieck form can be recasted as Kalman filtering on q-times integrated Wiener processes. Doing so provides a family of IVP solvers that return a Gaussian posterior measure, rather than a point estimate. We show that some such methods have low computational overhead, nontrivial convergence order, and that the posterior has a calibrated concentration rate. Additionally, we suggest a step size adaptation algorithm which completes the proposed method to a practically useful implementation, which we experimentally evaluate using a representative set of standard codes in the DETEST benchmark set.},
  langid = {english},
  file = {/home/steffen/OneDrive/Library/PhD/ProbablisticSolvers/Schober_et_al-2019-A_probabilistic_model_for_the_numerical_solution.pdf}
}

@unpublished{schoberProbabilisticODESolvers2014,
  title = {Probabilistic {{ODE Solvers}} with {{Runge-Kutta Means}}},
  author = {Schober, Michael and Duvenaud, David and Hennig, Philipp},
  date = {2014-10-24},
  eprint = {1406.2582},
  eprinttype = {arxiv},
  eprintclass = {cs, math, stat},
  url = {http://arxiv.org/abs/1406.2582},
  urldate = {2020-05-17},
  abstract = {Runge-Kutta methods are the classic family of solvers for ordinary differential equations (ODEs), and the basis for the state of the art. Like most numerical methods, they return point estimates. We construct a family of probabilistic numerical methods that instead return a Gauss-Markov process defining a probability distribution over the ODE solution. In contrast to prior work, we construct this family such that posterior means match the outputs of the Runge-Kutta family exactly, thus inheriting their proven good properties. Remaining degrees of freedom not identified by the match to Runge-Kutta are chosen such that the posterior probability measure fits the observed structure of the ODE. Our results shed light on the structure of Runge-Kutta solvers from a new direction, provide a richer, probabilistic output, have low computational cost, and raise new research questions.},
  langid = {english},
  keywords = {Computer Science - Machine Learning,Mathematics - Numerical Analysis,Statistics - Machine Learning},
  file = {/home/steffen/OneDrive/Library/PhD/ProbablisticSolvers/Schober_et_al-2014-Probabilistic_ODE_Solvers_with_Runge-Kutta_Means.pdf}
}

@unpublished{snelsonVariableNoiseDimensionality2012,
  title = {Variable Noise and Dimensionality Reduction for Sparse {{Gaussian}} Processes},
  author = {Snelson, Edward and Ghahramani, Zoubin},
  date = {2012-06-27},
  eprint = {1206.6873},
  eprinttype = {arxiv},
  eprintclass = {cs, stat},
  url = {http://arxiv.org/abs/1206.6873},
  urldate = {2021-02-19},
  abstract = {The sparse pseudo-input Gaussian process (SPGP) is a new approximation method for speeding up GP regression in the case of a large number of data points N. The approximation is controlled by the gradient optimization of a small set of M `pseudo-inputs', thereby reducing complexity from N\^3 to NM\^2. One limitation of the SPGP is that this optimization space becomes impractically big for high dimensional data sets. This paper addresses this limitation by performing automatic dimensionality reduction. A projection of the input space to a low dimensional space is learned in a supervised manner, alongside the pseudo-inputs, which now live in this reduced space. The paper also investigates the suitability of the SPGP for modeling data with input-dependent noise. A further extension of the model is made to make it even more powerful in this regard - we learn an uncertainty parameter for each pseudo-input. The combination of sparsity, reduced dimension, and input-dependent noise makes it possible to apply GPs to much larger and more complex data sets than was previously practical. We demonstrate the benefits of these methods on several synthetic and real world problems.},
  keywords = {Computer Science - Machine Learning,Statistics - Machine Learning},
  file = {/home/steffen/OneDrive/Library/PhD/Gaussian Processes/Sparse/Snelson_Ghahramani-2012-Variable_noise_and_dimensionality_reduction_for.pdf;/home/steffen/Zotero/storage/5WBD8KPU/1206.html}
}

@online{wilsonEfficientlySamplingFunctions2020,
  title = {Efficiently {{Sampling Functions}} from {{Gaussian Process Posteriors}}},
  author = {Wilson, James T. and Borovitskiy, Viacheslav and Terenin, Alexander and Mostowsky, Peter and Deisenroth, Marc Peter},
  date = {2020-08-16},
  eprint = {2002.09309},
  eprinttype = {arxiv},
  eprintclass = {cs, stat},
  url = {http://arxiv.org/abs/2002.09309},
  urldate = {2022-06-08},
  abstract = {Gaussian processes are the gold standard for many real-world modeling problems, especially in cases where a model’s success hinges upon its ability to faithfully represent predictive uncertainty. These problems typically exist as parts of larger frameworks, wherein quantities of interest are ultimately defined by integrating over posterior distributions. These quantities are frequently intractable, motivating the use of Monte Carlo methods. Despite substantial progress in scaling up Gaussian processes to large training sets, methods for accurately generating draws from their posterior distributions still scale cubically in the number of test locations. We identify a decomposition of Gaussian processes that naturally lends itself to scalable sampling by separating out the prior from the data. Building off of this factorization, we propose an easy-to-use and general-purpose approach for fast posterior sampling, which seamlessly pairs with sparse approximations to afford scalability both during training and at test time. In a series of experiments designed to test competing sampling schemes’ statistical properties and practical ramifications, we demonstrate how decoupled sample paths accurately represent Gaussian process posteriors at a fraction of the usual cost.},
  langid = {english},
  pubstate = {preprint},
  keywords = {Computer Science - Machine Learning,Statistics - Computation,Statistics - Machine Learning},
  file = {/home/steffen/OneDrive/Library/PhD/ProbablisticSolvers/Sampling/Wilson_et_al-2020-Efficiently_Sampling_Functions_from_Gaussian.pdf}
}
