@misc{gopakumar2024uncertaintyquantificationsurrogatemodels,
      title={Uncertainty Quantification of Surrogate Models using Conformal Prediction}, 
      author={Vignesh Gopakumar and Ander Gray and Joel Oskarsson and Lorenzo Zanisi and Stanislas Pamela and Daniel Giles and Matt Kusner and Marc Peter Deisenroth},
      year={2024},
      eprint={2408.09881},
      archivePrefix={arXiv},
      primaryClass={cs.AI},
      url={https://arxiv.org/abs/2408.09881}, 
}
@book{haykin1994neural,
  title={Neural Networks: A Comprehensive Foundation},
  author={Haykin, Simon},
  year={1994},
  publisher={Prentice Hall PTR}
}

@article{GOPAKUMAR2023100464,
title = {{Loss landscape engineering via data regulation on PINNs}},
journal = {Machine Learning with Applications},
volume = {12},
pages = {100464},
year = {2023},
issn = {2666-8270},
doi = {https://doi.org/10.1016/j.mlwa.2023.100464},
url = {https://www.sciencedirect.com/science/article/pii/S2666827023000178},
author = {Vignesh Gopakumar and Stanislas Pamela and Debasmita Samaddar},
keywords = {Physics-Informed Neural Networks, Loss landscape, Sparse regularisation, Partial differential equations, Optimisation},
abstract = {Physics-Informed Neural Networks have shown unique utility in parameterising the solution of a well-defined partial differential equation using automatic differentiation and residual losses. Though they provide theoretical guarantees of convergence, in practice the required training regimes tend to be exacting and demanding. Through the course of this paper, we take a deep dive into understanding the loss landscapes associated with a PINN and how that offers some insight as to why PINNs are fundamentally hard to optimise for. We demonstrate how PINNs can be forced to converge better towards the solution, by way of feeding in sparse or coarse data as a regulator. The data regulates and morphs the topology of the loss landscape associated with the PINN to make it easily traversable for the minimiser. Data regulation of PINNs helps ease the optimisation required for convergence by invoking a hybrid unsupervised–supervised training approach, where the labelled data pushes the network towards the vicinity of the solution, and the unlabelled regime fine-tunes it to the solution.}
}

@misc{brundage2020toward,
      title={{Toward Trustworthy AI Development: Mechanisms for Supporting Verifiable Claims}}, 
      author={Brundage, Miles and Avin, Shahar and Wang, Jasmine and Belfield, Haydn and Krueger, Gretchen and Hadfield, Gillian and Khlaaf, Heidy and Yang, Jingying and Toner, Helen and Fong, Ruth and others},
      year={2020},
      eprint={2004.07213},
      archivePrefix={arXiv},
      primaryClass={cs.CY},
      url={https://arxiv.org/abs/2004.07213}, 
}
@article{balch2019satellite,
  title={Satellite conjunction analysis and the false confidence theorem},
  author={Balch, Michael Scott and Martin, Ryan and Ferson, Scott},
  journal={Proceedings of the Royal Society A},
  volume={475},
  number={2227},
  pages={20180565},
  year={2019},
  publisher={The Royal Society Publishing}
}
@article{lazysets21,
  title     = {{LazySets.jl: Scalable Symbolic-Numeric Set Computations}},
  author    = {Forets, Marcelo and Schilling, Christian},
  journal   = {Proceedings of the JuliaCon Conferences},
  year      = {2021},
  publisher = {The Open Journal},
  volume    = {1},
  number    = {1},
  pages     = {11},
  doi       = {10.21105/jcon.00097}
}
@article{benet2019taylorseries,
  title={{TaylorSeries.jl}: {T}aylor expansions in one and several variables in {J}ulia},
  author={Benet, Luis and Sanders, David P},
  journal={Journal of Open Source Software},
  volume={4},
  number={36},
  pages={1043},
  year={2019}
}
@article{messoudi2021copula,
  title={Copula-based conformal prediction for multi-target regression},
  author={Messoudi, Soundouss and Destercke, S{\'e}bastien and Rousseau, Sylvain},
  journal={Pattern Recognition},
  volume={120},
  pages={108101},
  year={2021},
  publisher={Elsevier}
}
@inproceedings{sun2023copula,
  title={Copula Conformal prediction for multi-step time series prediction},
  author={Sun, Sophia Huiwen and Yu, Rose},
  booktitle={The Twelfth International Conference on Learning Representations},
  year={2023}
}
@article{diquigiovanni2022conformal,
  title={Conformal prediction bands for multivariate functional data},
  author={Diquigiovanni, Jacopo and Fontana, Matteo and Vantini, Simone},
  journal={Journal of Multivariate Analysis},
  volume={189},
  pages={104879},
  year={2022},
  publisher={Elsevier}
}
@article{li2020fourier,
  title={{F}ourier neural operator for parametric partial differential equations},
  author={Li, Zongyi and Kovachki, Nikola and Azizzadenesheli, Kamyar and Liu, Burigede and Bhattacharya, Kaushik and Stuart, Andrew and Anandkumar, Anima},
  journal={arXiv preprint arXiv:2010.08895},
  year={2020}
}
@article{gopakumar2024plasma,
  title={Plasma surrogate modelling using {F}ourier neural operators},
  author={Gopakumar, Vignesh and Pamela, Stanislas and Zanisi, Lorenzo and Li, Zongyi and Gray, Ander and Brennand, Daniel and Bhatia, Nitesh and Stathopoulos, Gregory and Kusner, Matt and Deisenroth, Marc Peter and others},
  journal={Nuclear Fusion},
  volume={64},
  number={5},
  pages={056025},
  year={2024},
  publisher={IOP Publishing}
}
@article{azizzadenesheli2024neural,
  title={Neural operators for accelerating scientific simulations and design},
  author={Azizzadenesheli, Kamyar and Kovachki, Nikola and Li, Zongyi and Liu-Schiaffini, Miguel and Kossaifi, Jean and Anandkumar, Anima},
  journal={Nature Reviews Physics},
  pages={1--9},
  year={2024},
  publisher={Nature Publishing Group UK London}
}
@inproceedings{kurth2023fourcastnet,
  title={Fourcastnet: Accelerating global high-resolution weather forecasting using adaptive {F}ourier neural operators},
  author={Kurth, Thorsten and Subramanian, Shashank and Harrington, Peter and Pathak, Jaideep and Mardani, Morteza and Hall, David and Miele, Andrea and Kashinath, Karthik and Anandkumar, Anima},
  booktitle={Proceedings of the platform for advanced scientific computing conference},
  pages={1--11},
  year={2023}
}
@inproceedings{messoudi2022ellipsoidal,
  title={Ellipsoidal conformal inference for multi-target regression},
  author={Messoudi, Soundouss and Destercke, S{\'e}bastien and Rousseau, Sylvain},
  booktitle={Conformal and Probabilistic Prediction with Applications},
  pages={294--306},
  year={2022},
  organization={PMLR}
}
@inproceedings{de2021constructing,
  title={Constructing consonant predictive beliefs from data with scenario theory},
  author={De Angelis, Marco and Rocchetta, Roberto and Gray, Ander and Ferson, Scott},
  booktitle={International Symposium on Imprecise Probability: Theories and Applications},
  pages={357--360},
  year={2021},
  organization={PMLR}
}
@book{vovk2005algorithmic,
  title={Algorithmic learning in a random world},
  author={Vovk, Vladimir and Gammerman, Alexander and Shafer, Glenn},
  volume={29},
  year={2005},
  publisher={Springer}
}
@article{cella2022validity,
  title={Validity, consonant plausibility measures, and conformal prediction},
  author={Cella, Leonardo and Martin, Ryan},
  journal={International Journal of Approximate Reasoning},
  volume={141},
  pages={110--130},
  year={2022},
  publisher={Elsevier}
}
@book{shafer1976mathematical,
  title={A mathematical theory of evidence},
  author={Shafer, Glenn},
  volume={42},
  year={1976},
  publisher={Princeton university press}
}
@book{martin2015inferential,
  title={Inferential models: reasoning with uncertainty},
  author={Martin, Ryan and Liu, Chuanhai},
  year={2015},
  publisher={CRC Press}
}
@article{denoeux2018frequency,
  title={Frequency-calibrated belief functions: review and new insights},
  author={Den{\oe}ux, Thierry and Li, Shoumei},
  journal={International Journal of Approximate Reasoning},
  volume={92},
  pages={232--254},
  year={2018},
  publisher={Elsevier}
}
@article{balch2012mathematical,
  title={Mathematical foundations for a theory of confidence structures},
  author={Balch, Michael Scott},
  journal={International Journal of Approximate Reasoning},
  volume={53},
  number={7},
  pages={1003--1019},
  year={2012},
  publisher={Elsevier}
}
@article{dalrymple2024towards,
  title={{Towards Guaranteed Safe AI: A Framework for Ensuring Robust and Reliable AI Systems}},
  author={Dalrymple, David and Skalse, Joar and Bengio, Yoshua and Russell, Stuart and Tegmark, Max and Seshia, Sanjit and Omohundro, Steve and Szegedy, Christian and Goldhaber, Ben and Ammann, Nora and others},
  journal={CoRR},
  year={2024}
}
@inproceedings{papadopoulos2002inductive,
  title={Inductive confidence machines for regression},
  author={Papadopoulos, Harris and Proedrou, Kostas and Vovk, Volodya and Gammerman, Alex},
  booktitle={Machine learning: ECML 2002: 13th European conference on machine learning Helsinki, Finland, August 19--23, 2002 proceedings 13},
  pages={345--356},
  year={2002},
  organization={Springer}
}
@article{GUPTA2022108496,
title = {Nested conformal prediction and quantile out-of-bag ensemble methods},
journal = {Pattern Recognition},
volume = {127},
pages = {108496},
year = {2022},
issn = {0031-3203},
doi = {https://doi.org/10.1016/j.patcog.2021.108496},
url = {https://www.sciencedirect.com/science/article/pii/S0031320321006725},
author = {Chirag Gupta and Arun K. Kuchibhotla and Aaditya Ramdas},
keywords = {Conformal prediction, Quantile regression, Cross-conformal, Out-of-bag methods, Ensemble methods, Random forests},
abstract = {Conformal prediction is a popular tool for providing valid prediction sets for classification and regression problems, without relying on any distributional assumptions on the data. While the traditional description of conformal prediction starts with a nonconformity score, we provide an alternate (but equivalent) view that starts with a sequence of nested sets and calibrates them to find a valid prediction set. The nested framework subsumes all nonconformity scores, including recent proposals based on quantile regression and density estimation. While these ideas were originally derived based on sample splitting, our framework seamlessly extends them to other aggregation schemes like cross-conformal, jackknife+ and out-of-bag methods. We use the framework to derive a new algorithm (QOOB, pronounced cube) that combines four ideas: quantile regression, cross-conformalization, ensemble methods and out-of-bag predictions. We develop a computationally efficient implementation of cross-conformal, that is also used by QOOB. In a detailed numerical investigation, QOOB performs either the best or close to the best on all simulated and real datasets.}
}
@book{canuto2007spectral,
  title={Spectral Methods: Evolution to Complex Geometries and Applications to Fluid Dynamics},
  author={Canuto, C},
  year={2007},
  publisher={Springer-Verlag}
}
@inproceedings{guibas2003zonotopes,
  title={Zonotopes as bounding volumes.},
  author={Guibas, Leonidas J and Nguyen, An Thanh and Zhang, Li},
  booktitle={SODA},
  volume={3},
  pages={803--812},
  year={2003},
  organization={Citeseer}
}
@article{
ma2024calibrated,
title={Calibrated Uncertainty Quantification for Operator Learning via Conformal Prediction},
author={Ziqi Ma and David Pitt and Kamyar Azizzadenesheli and Anima Anandkumar},
journal={Transactions on Machine Learning Research},
issn={2835-8856},
year={2024},
url={https://openreview.net/forum?id=cGpegxy12T},
note={}
}

@thesis{pal2023efficient,
  title     = {{On Efficient Training \& Inference of Neural Differential Equations}},
  author    = {Pal, Avik},
  year      = {2023},
  school    = {Massachusetts Institute of Technology}
}

@article{redmond2009communities,
  title={Communities and crime data set},
  author={Redmond, Michael and Baveja, A},
  journal={UCI Machine Learning Repository},
  year={2009},
  publisher={University of California, School of Information and Computer Science}
}
@inproceedings{nugteren2015cltune,
  title={CLTune: A generic auto-tuner for OpenCL kernels},
  author={Nugteren, Cedric and Codreanu, Valeriu},
  booktitle={2015 IEEE 9th International Symposium on Embedded Multicore/Many-core Systems-on-Chip},
  pages={195--202},
  year={2015},
  organization={IEEE}
}
@inproceedings{torres2014ujiindoorloc,
  title={UJIIndoorLoc: A new multi-building and multi-floor database for WLAN fingerprint-based indoor localization problems},
  author={Torres-Sospedra, Joaqu{\'\i}n and Montoliu, Ra{\'u}l and Mart{\'\i}nez-Us{\'o}, Adolfo and Avariento, Joan P and Arnau, Tom{\'a}s J and Benedito-Bordonau, Mauri and Huerta, Joaqu{\'\i}n},
  booktitle={2014 international conference on indoor positioning and indoor navigation (IPIN)},
  pages={261--270},
  year={2014},
  organization={IEEE}
}
@inproceedings{zhou2014predicting,
  title={Predicting the geographical origin of music},
  author={Zhou, Fang and Claire, Q and King, Ross D},
  booktitle={2014 IEEE International Conference on Data Mining},
  pages={1115--1120},
  year={2014},
  organization={IEEE}
}
@article{cho2020comparative,
  title={Comparative assessment of various machine learning-based bias correction methods for numerical weather prediction model forecasts of extreme air temperatures in urban areas},
  author={Cho, Dongjin and Yoo, Cheolhee and Im, Jungho and Cha, Dong-Hyun},
  journal={Earth and Space Science},
  volume={7},
  number={4},
  pages={e2019EA000740},
  year={2020},
  publisher={Wiley Online Library}
}
@article{YANG2022115399,
title = {Scalable uncertainty quantification for deep operator networks using randomized priors},
journal = {Computer Methods in Applied Mechanics and Engineering},
volume = {399},
pages = {115399},
year = {2022},
issn = {0045-7825},
doi = {https://doi.org/10.1016/j.cma.2022.115399},
url = {https://www.sciencedirect.com/science/article/pii/S0045782522004595},
author = {Yibo Yang and Georgios Kissas and Paris Perdikaris},
keywords = {Deep learning, Uncertainty quantification, Operator learning},
abstract = {We present a simple and effective approach for posterior uncertainty quantification in deep operator networks (DeepONets); an emerging paradigm for supervised learning in function spaces. We adopt a frequentist approach based on randomized prior ensembles, and put forth an efficient vectorized implementation for fast parallel inference on accelerated hardware. Through a collection of representative examples in computational mechanics and climate modeling, we show that the merits of the proposed approach are fourfold. (1) It can provide more robust and accurate predictions when compared against deterministic DeepONets. (2) It shows great capability in providing reliable uncertainty estimates on scarce data sets with multi-scale function pairs. (3) It can effectively detect out-of-distribution and adversarial examples. (4) It can seamlessly quantify uncertainty due to model bias, as well as noise corruption in the data. Finally, we provide an optimized JAX library called UQDeepONet that can accommodate large model architectures, large ensemble sizes, as well as large data sets with excellent parallel performance on accelerated hardware, thereby enabling uncertainty quantification for DeepONets in realistic large-scale applications. All code and data accompanying this manuscript will be made available at https://github.com/PredictiveIntelligenceLab/UQDeepONet.}
}

@inproceedings{beltran2024galerkin,
  title={Galerkin meets Laplace: Fast uncertainty estimation in neural PDEs},
  author={Beltran, Christian Jimenez and Vergari, Antonio and Teckentrup, Aretha L and Zygalakis, Konstantinos C},
  booktitle={ICLR 2024 Workshop on AI4DifferentialEquations In Science},
  year={2024}
}
@article{yang2019adversarial,
  title={Adversarial uncertainty quantification in physics-informed neural networks},
  author={Yang, Yibo and Perdikaris, Paris},
  journal={Journal of Computational Physics},
  volume={394},
  pages={136--152},
  year={2019},
  publisher={Elsevier}
}
@article{DUBOIS1990419,
title = {Consonant approximations of belief functions},
journal = {International Journal of Approximate Reasoning},
volume = {4},
number = {5},
pages = {419-449},
year = {1990},
issn = {0888-613X},
doi = {https://doi.org/10.1016/0888-613X(90)90015-T},
url = {https://www.sciencedirect.com/science/article/pii/0888613X9090015T},
author = {Didier Dubois and Henri Prade},
keywords = {belief functions, random sets, possibility theory},
abstract = {A general notion of approximation of a belief function by some other set function is introduced that is based on a recently introduced definition of inclusion between random sets. Viewing a fuzzy set as a consonant random set, it is shown how to construct fuzzy sets that may act as approximations of belief functions. Two kinds of approximations are considered: inner approximations that provide upper bounds on belief degrees and lower bounds on plausibility degrees, and outer approximations that provide lower bounds on belief degrees and upper bounds on plausibility degrees. Minimal outer and maximal inner consonant approximations are characterized in a constructive way. The particular problem of approximating a probability measure by a fuzzy set is solved. Applications to the approximate computation of belief functions on Cartesian products, combinations by Dempster's rule, and functions of random-set-valued arguments by means of fuzzy set operations are sketched.}
}
@article{shafer_CP,
author = {Shafer, Glenn and Vovk, Vladimir},
title = {A Tutorial on Conformal Prediction},
year = {2008},
issue_date = {6/1/2008},
publisher = {JMLR.org},
volume = {9},
issn = {1532-4435},
abstract = {Conformal prediction uses past experience to determine precise levels of confidence in new predictions. Given an error probability ε, together with a method that makes a prediction undefined of a label y, it produces a set of labels, typically containing undefined, that also contains y with probability 1 – ε. Conformal prediction can be applied to any method for producing undefined: a nearest-neighbor method, a support-vector machine, ridge regression, etc.Conformal prediction is designed for an on-line setting in which labels are predicted successively, each one being revealed before the next is predicted. The most novel and valuable feature of conformal prediction is that if the successive examples are sampled independently from the same distribution, then the successive predictions will be right 1 – ε of the time, even though they are based on an accumulating data set rather than on independent data sets.In addition to the model under which successive examples are sampled independently, other on-line compression models can also use conformal prediction. The widely used Gaussian linear model is one of these.This tutorial presents a self-contained account of the theory of conformal prediction and works through several numerical examples. A more comprehensive treatment of the topic is provided in Algorithmic Learning in a Random World, by Vladimir Vovk, Alex Gammerman, and Glenn Shafer (Springer, 2005).},
journal = {J. Mach. Learn. Res.},
month = jun,
pages = {371–421},
numpages = {51}
}
@article{plassier2025rectifying,
  title={Rectifying Conformity Scores for Better Conditional Coverage},
  author={Plassier, Vincent and Fishkov, Alexander and Dheur, Victor and Guizani, Mohsen and Taieb, Souhaib Ben and Panov, Maxim and Moulines, Eric},
  journal={CoRR},
  year={2025}
}