# XNet-Enhanced Deep BSDE Method and Numerical Analysis

arXiv: https://arxiv.org/abs/2502.06238 · HTML: https://mlsanity.com/paper/2502.06238/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Feb 2025.

## Built on by (most cited first, 1 of 1)

- [A Reverse-BSDE Diffusion Sampler](https://mlsanity.com/paper/2505.06800/) (2025)

## Abstract (arXiv)

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.