XNet-Enhanced Deep BSDE Method and Numerical Analysis

TL;DR

This paper advances Deep BSDE methods by establishing convergence for non-Lipschitz generators and introduces XNet, a computationally efficient architecture, validated through high-dimensional PDE experiments.

Contribution

It provides the first convergence theory for non-Lipschitz generators in Deep BSDE methods and proposes XNet, a shallow architecture that reduces computational costs.

Findings

Convergence established for Allen--Cahn and HJB equations with non-Lipschitz generators.

XNet architecture achieves strong approximation with fewer parameters.

Numerical experiments show efficiency gains in 100-dimensional PDEs.

Abstract

Semilinear parabolic partial differential equations (PDEs) are fundamental to modeling complex dynamical systems across scientific domains. The Deep Backward Stochastic Differential Equation (BSDE) method is a promising approach for high-dimensional PDEs; however, existing convergence results apply only to globally Lipschitz generators, excluding important cases such as Allen--Cahn and Hamilton--Jacobi--Bellman (HJB) equations. This paper presents both a theoretical and a computational advance for Deep BSDE methods. Theoretically, we establish the convergence theory for non--Lipschitz generators--covering Allen--Cahn equations with cubic nonlinearity and HJB equations with quadratic gradient growth--based on a bounded double--well lemma and a truncated-BSDE analysis within the Bouchard--Touzi--Zhang theory. Computationally, we instantiate the framework with XNet, a shallow architecture with $\mathcal O(L)$ parameters that preserves strong approximation while substantially reducing optimization and computational cost. Numerical experiments on 100--dimensional PDEs corroborate the predicted convergence behavior and demonstrate significant efficiency gains over standard feedforward implementations.