Unbiased and Second-Order-Free Training for High-Dimensional PDEs

TL;DR

This paper introduces an unbiased, second-order-free training framework for high-dimensional PDEs using BSDEs, avoiding the bias of Euler-Maruyama discretization while maintaining computational efficiency.

Contribution

It provides a principled analysis of EM-induced bias and proposes a novel training method that eliminates bias without second-order derivatives, with code available online.

Findings

The proposed method removes bias caused by Euler-Maruyama discretization.

It maintains the computational simplicity of BSDE approaches.

The code implementation is publicly available.

Abstract

Deep learning methods based on backward stochastic differential equations (BSDEs) have emerged as competitive alternatives to physics-informed neural networks (PINNs) for solving high-dimensional partial differential equations (PDEs). By leveraging probabilistic representations, BSDE approaches can avoid the curse of dimensionality and often admit second-order-free training objectives that do not require explicit Hessian evaluations. It has recently been established that the commonly used Euler-Maruyama (EM) time discretization induces an intrinsic bias in BSDE training losses. While high-order schemes such as Heun can fully eliminate this bias, such schemes re-introduce second-order spatial derivatives and incur substantial computational overhead. In this work, we provide a principled analysis of EM-induced loss bias and propose an unbiased, second-order-free training framework that preserves the computational advantages of BSDE methods. Our code is available at https://github.com/seojaemin22/Un-EM-BSDE.