Generalization Error Analysis of Deep Backward Dynamic Programming for Solving Nonlinear PDEs

TL;DR

This paper analyzes how quasi-Monte Carlo methods improve the convergence rate of generalization error in deep backward dynamic programming for solving high-dimensional nonlinear PDEs, outperforming traditional Monte Carlo methods.

Contribution

It provides a theoretical and empirical comparison showing QMC methods achieve faster convergence of generalization error than Monte Carlo in deep PDE solvers.

Findings

QMC methods have a convergence rate of O(m^{-1+ε}) for generalization error.

MC methods have a convergence rate of O(m^{-1/2+ε}) for generalization error.

Numerical experiments confirm QMC's superior accuracy and stability.

Abstract

We explore the application of the quasi-Monte Carlo (QMC) method in deep backward dynamic programming (DBDP) (Hure et al. 2020) for numerically solving high-dimensional nonlinear partial differential equations (PDEs). Our study focuses on examining the generalization error as a component of the total error in the DBDP framework, discovering that the rate of convergence for the generalization error is influenced by the choice of sampling methods. Specifically, for a given batch size $m$, the generalization error under QMC methods exhibits a convergence rate of $O(m^{-1+\varepsilon})$, where $\varepsilon$ can be made arbitrarily small. This rate is notably more favorable than that of the traditional Monte Carlo (MC) methods, which is $O(m^{-1/2+\varepsilon})$. Our theoretical analysis shows that the generalization error under QMC methods achieves a higher order of convergence than their MC counterparts. Numerical experiments demonstrate that QMC indeed surpasses MC in delivering solutions that are both more precise and stable.