Probabilistic Numerical Method of Lines for Time-Dependent Partial Differential Equations

Nicholas Krämer, Jonathan Schmidt, Philipp Hennig
Proceedings of The 25th International Conference on Artificial Intelligence and Statistics, PMLR 151:625-639, 2022.

Abstract

This work develops a class of probabilistic algorithms for the numerical solution of nonlinear, time-dependent partial differential equations (PDEs). Current state-of-the-art PDE solvers treat the space- and time-dimensions separately, serially, and with black-box algorithms, which obscures the interactions between spatial and temporal approximation errors and misguides the quantification of the overall error. To fix this issue, we introduce a probabilistic version of a technique called method of lines. The proposed algorithm begins with a Gaussian process interpretation of finite difference methods, which then interacts naturally with filtering-based probabilistic ordinary differential equation (ODE) solvers because they share a common language: Bayesian inference. Joint quantification of space- and time-uncertainty becomes possible without losing the performance benefits of well-tuned ODE solvers. Thereby, we extend the toolbox of probabilistic programs for differential equation simulation to PDEs.

Cite this Paper

BibTeX
@InProceedings{pmlr-v151-kramer22a, title = { Probabilistic Numerical Method of Lines for Time-Dependent Partial Differential Equations }, author = {Kr\"amer, Nicholas and Schmidt, Jonathan and Hennig, Philipp}, booktitle = {Proceedings of The 25th International Conference on Artificial Intelligence and Statistics}, pages = {625--639}, year = {2022}, editor = {Camps-Valls, Gustau and Ruiz, Francisco J. R. and Valera, Isabel}, volume = {151}, series = {Proceedings of Machine Learning Research}, month = {28--30 Mar}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v151/kramer22a/kramer22a.pdf}, url = {https://proceedings.mlr.press/v151/kramer22a.html}, abstract = { This work develops a class of probabilistic algorithms for the numerical solution of nonlinear, time-dependent partial differential equations (PDEs). Current state-of-the-art PDE solvers treat the space- and time-dimensions separately, serially, and with black-box algorithms, which obscures the interactions between spatial and temporal approximation errors and misguides the quantification of the overall error. To fix this issue, we introduce a probabilistic version of a technique called method of lines. The proposed algorithm begins with a Gaussian process interpretation of finite difference methods, which then interacts naturally with filtering-based probabilistic ordinary differential equation (ODE) solvers because they share a common language: Bayesian inference. Joint quantification of space- and time-uncertainty becomes possible without losing the performance benefits of well-tuned ODE solvers. Thereby, we extend the toolbox of probabilistic programs for differential equation simulation to PDEs. } }
Endnote
%0 Conference Paper %T Probabilistic Numerical Method of Lines for Time-Dependent Partial Differential Equations %A Nicholas Krämer %A Jonathan Schmidt %A Philipp Hennig %B Proceedings of The 25th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2022 %E Gustau Camps-Valls %E Francisco J. R. Ruiz %E Isabel Valera %F pmlr-v151-kramer22a %I PMLR %P 625--639 %U https://proceedings.mlr.press/v151/kramer22a.html %V 151 %X This work develops a class of probabilistic algorithms for the numerical solution of nonlinear, time-dependent partial differential equations (PDEs). Current state-of-the-art PDE solvers treat the space- and time-dimensions separately, serially, and with black-box algorithms, which obscures the interactions between spatial and temporal approximation errors and misguides the quantification of the overall error. To fix this issue, we introduce a probabilistic version of a technique called method of lines. The proposed algorithm begins with a Gaussian process interpretation of finite difference methods, which then interacts naturally with filtering-based probabilistic ordinary differential equation (ODE) solvers because they share a common language: Bayesian inference. Joint quantification of space- and time-uncertainty becomes possible without losing the performance benefits of well-tuned ODE solvers. Thereby, we extend the toolbox of probabilistic programs for differential equation simulation to PDEs.
APA
Krämer, N., Schmidt, J. & Hennig, P.. (2022). Probabilistic Numerical Method of Lines for Time-Dependent Partial Differential Equations . Proceedings of The 25th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 151:625-639 Available from https://proceedings.mlr.press/v151/kramer22a.html.

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