Gaussian Process Priors for Systems of Linear Partial Differential Equations with Constant Coefficients

Marc Harkonen, Markus Lange-Hegermann, Bogdan Raita
Proceedings of the 40th International Conference on Machine Learning, PMLR 202:12587-12615, 2023.

Abstract

Partial differential equations (PDEs) are important tools to model physical systems and including them into machine learning models is an important way of incorporating physical knowledge. Given any system of linear PDEs with constant coefficients, we propose a family of Gaussian process (GP) priors, which we call EPGP, such that all realizations are exact solutions of this system. We apply the Ehrenpreis-Palamodov fundamental principle, which works as a non-linear Fourier transform, to construct GP kernels mirroring standard spectral methods for GPs. Our approach can infer probable solutions of linear PDE systems from any data such as noisy measurements, or pointwise defined initial and boundary conditions. Constructing EPGP-priors is algorithmic, generally applicable, and comes with a sparse version (S-EPGP) that learns the relevant spectral frequencies and works better for big data sets. We demonstrate our approach on three families of systems of PDEs, the heat equation, wave equation, and Maxwell’s equations, where we improve upon the state of the art in computation time and precision, in some experiments by several orders of magnitude.

Cite this Paper

BibTeX
@InProceedings{pmlr-v202-harkonen23a, title = {{G}aussian Process Priors for Systems of Linear Partial Differential Equations with Constant Coefficients}, author = {Harkonen, Marc and Lange-Hegermann, Markus and Raita, Bogdan}, booktitle = {Proceedings of the 40th International Conference on Machine Learning}, pages = {12587--12615}, year = {2023}, editor = {Krause, Andreas and Brunskill, Emma and Cho, Kyunghyun and Engelhardt, Barbara and Sabato, Sivan and Scarlett, Jonathan}, volume = {202}, series = {Proceedings of Machine Learning Research}, month = {23--29 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v202/harkonen23a/harkonen23a.pdf}, url = {https://proceedings.mlr.press/v202/harkonen23a.html}, abstract = {Partial differential equations (PDEs) are important tools to model physical systems and including them into machine learning models is an important way of incorporating physical knowledge. Given any system of linear PDEs with constant coefficients, we propose a family of Gaussian process (GP) priors, which we call EPGP, such that all realizations are exact solutions of this system. We apply the Ehrenpreis-Palamodov fundamental principle, which works as a non-linear Fourier transform, to construct GP kernels mirroring standard spectral methods for GPs. Our approach can infer probable solutions of linear PDE systems from any data such as noisy measurements, or pointwise defined initial and boundary conditions. Constructing EPGP-priors is algorithmic, generally applicable, and comes with a sparse version (S-EPGP) that learns the relevant spectral frequencies and works better for big data sets. We demonstrate our approach on three families of systems of PDEs, the heat equation, wave equation, and Maxwell’s equations, where we improve upon the state of the art in computation time and precision, in some experiments by several orders of magnitude.} }
Endnote
%0 Conference Paper %T Gaussian Process Priors for Systems of Linear Partial Differential Equations with Constant Coefficients %A Marc Harkonen %A Markus Lange-Hegermann %A Bogdan Raita %B Proceedings of the 40th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2023 %E Andreas Krause %E Emma Brunskill %E Kyunghyun Cho %E Barbara Engelhardt %E Sivan Sabato %E Jonathan Scarlett %F pmlr-v202-harkonen23a %I PMLR %P 12587--12615 %U https://proceedings.mlr.press/v202/harkonen23a.html %V 202 %X Partial differential equations (PDEs) are important tools to model physical systems and including them into machine learning models is an important way of incorporating physical knowledge. Given any system of linear PDEs with constant coefficients, we propose a family of Gaussian process (GP) priors, which we call EPGP, such that all realizations are exact solutions of this system. We apply the Ehrenpreis-Palamodov fundamental principle, which works as a non-linear Fourier transform, to construct GP kernels mirroring standard spectral methods for GPs. Our approach can infer probable solutions of linear PDE systems from any data such as noisy measurements, or pointwise defined initial and boundary conditions. Constructing EPGP-priors is algorithmic, generally applicable, and comes with a sparse version (S-EPGP) that learns the relevant spectral frequencies and works better for big data sets. We demonstrate our approach on three families of systems of PDEs, the heat equation, wave equation, and Maxwell’s equations, where we improve upon the state of the art in computation time and precision, in some experiments by several orders of magnitude.
APA
Harkonen, M., Lange-Hegermann, M. & Raita, B.. (2023). Gaussian Process Priors for Systems of Linear Partial Differential Equations with Constant Coefficients. Proceedings of the 40th International Conference on Machine Learning, in Proceedings of Machine Learning Research 202:12587-12615 Available from https://proceedings.mlr.press/v202/harkonen23a.html.

Related Material