Robust SDE-Based Variational Formulations for Solving Linear PDEs via Deep Learning
This work addresses the challenge of efficient and accurate PDE solving for high-dimensional applications, though it appears incremental as it builds on existing deep learning and Monte Carlo approaches.
The paper tackled the problem of solving linear Kolmogorov PDEs in high dimensions using SDE-based variational formulations with deep learning, by systematically comparing existing methods and proposing novel ones that are more robust, leading to substantial performance improvements.
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.