Robust SDE-Based Variational Formulations for Solving Linear PDEs via Deep Learning

Lorenz Richter, Julius Berner
Proceedings of the 39th International Conference on Machine Learning, PMLR 162:18649-18666, 2022.

Abstract

The combination of Monte Carlo methods and deep learning has recently led to efficient algorithms for solving partial differential equations (PDEs) in high dimensions. Related learning problems are often stated as variational formulations based on associated stochastic differential equations (SDEs), which allow the minimization of corresponding losses using gradient-based optimization methods. In respective numerical implementations it is therefore crucial to rely on adequate gradient estimators that exhibit low variance in order to reach convergence accurately and swiftly. In this article, we rigorously investigate corresponding numerical aspects that appear in the context of linear Kolmogorov PDEs. In particular, we systematically compare existing deep learning approaches and provide theoretical explanations for their performances. Subsequently, we suggest novel methods that can be shown to be more robust both theoretically and numerically, leading to substantial performance improvements.

Cite this Paper

BibTeX
@InProceedings{pmlr-v162-richter22a, title = {Robust {SDE}-Based Variational Formulations for Solving Linear {PDE}s via Deep Learning}, author = {Richter, Lorenz and Berner, Julius}, booktitle = {Proceedings of the 39th International Conference on Machine Learning}, pages = {18649--18666}, year = {2022}, editor = {Chaudhuri, Kamalika and Jegelka, Stefanie and Song, Le and Szepesvari, Csaba and Niu, Gang and Sabato, Sivan}, volume = {162}, series = {Proceedings of Machine Learning Research}, month = {17--23 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v162/richter22a/richter22a.pdf}, url = {https://proceedings.mlr.press/v162/richter22a.html}, abstract = {The combination of Monte Carlo methods and deep learning has recently led to efficient algorithms for solving partial differential equations (PDEs) in high dimensions. Related learning problems are often stated as variational formulations based on associated stochastic differential equations (SDEs), which allow the minimization of corresponding losses using gradient-based optimization methods. In respective numerical implementations it is therefore crucial to rely on adequate gradient estimators that exhibit low variance in order to reach convergence accurately and swiftly. In this article, we rigorously investigate corresponding numerical aspects that appear in the context of linear Kolmogorov PDEs. In particular, we systematically compare existing deep learning approaches and provide theoretical explanations for their performances. Subsequently, we suggest novel methods that can be shown to be more robust both theoretically and numerically, leading to substantial performance improvements.} }
Endnote
%0 Conference Paper %T Robust SDE-Based Variational Formulations for Solving Linear PDEs via Deep Learning %A Lorenz Richter %A Julius Berner %B Proceedings of the 39th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2022 %E Kamalika Chaudhuri %E Stefanie Jegelka %E Le Song %E Csaba Szepesvari %E Gang Niu %E Sivan Sabato %F pmlr-v162-richter22a %I PMLR %P 18649--18666 %U https://proceedings.mlr.press/v162/richter22a.html %V 162 %X The combination of Monte Carlo methods and deep learning has recently led to efficient algorithms for solving partial differential equations (PDEs) in high dimensions. Related learning problems are often stated as variational formulations based on associated stochastic differential equations (SDEs), which allow the minimization of corresponding losses using gradient-based optimization methods. In respective numerical implementations it is therefore crucial to rely on adequate gradient estimators that exhibit low variance in order to reach convergence accurately and swiftly. In this article, we rigorously investigate corresponding numerical aspects that appear in the context of linear Kolmogorov PDEs. In particular, we systematically compare existing deep learning approaches and provide theoretical explanations for their performances. Subsequently, we suggest novel methods that can be shown to be more robust both theoretically and numerically, leading to substantial performance improvements.
APA
Richter, L. & Berner, J.. (2022). Robust SDE-Based Variational Formulations for Solving Linear PDEs via Deep Learning. Proceedings of the 39th International Conference on Machine Learning, in Proceedings of Machine Learning Research 162:18649-18666 Available from https://proceedings.mlr.press/v162/richter22a.html.

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