Unbiased and Second-Order-Free Training for High-Dimensional PDEs
For researchers using BSDE methods to solve high-dimensional PDEs, this work removes a previously unrecognized bias while preserving computational advantages.
The authors identify and analyze the bias introduced by Euler-Maruyama discretization in BSDE-based PDE solvers, and propose a training framework that eliminates this bias without requiring second-order derivatives, maintaining computational efficiency.
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.