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

2026-05-14

AI Summary

This work addresses the bias in training loss introduced by Euler–Maruyama time discretization in deep learning methods based on backward stochastic differential equations (BSDEs) for solving high-dimensional partial differential equations (PDEs). Through a theoretical analysis of this discretization error, the authors propose a novel unbiased training framework that eliminates the need to compute second-order spatial derivatives. This approach yields the first unbiased BSDE-based learning algorithm that completely avoids Hessian computations, thereby achieving significantly improved solution accuracy while maintaining high computational efficiency. The method is specifically designed for high-dimensional PDEs and is accompanied by publicly released code.

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.

Problem

• high-dimensional PDEs
• backward stochastic differential equations
• Euler-Maruyama discretization
• training bias
• second-order derivatives

Innovation

• unbiased training
• second-order-free
• backward stochastic differential equations
• high-dimensional PDEs
• Euler-Maruyama bias