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

TL;DR

This paper develops robust stochastic differential equation-based variational methods for solving linear PDEs with deep learning, improving convergence and performance through theoretical analysis and novel algorithms.

Contribution

It introduces new SDE-based variational formulations for linear PDEs, offering enhanced robustness and efficiency over existing methods.

Findings

New methods exhibit lower variance in gradient estimators.

Theoretical analysis explains performance differences among approaches.

Numerical experiments demonstrate substantial performance improvements.

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.