提出无偏且无需二阶导数的高维偏微分方程求解方法
Unbiased and Second-Order-Free Training for High-Dimensional PDEs
- 基于随机微分方程的无偏训练框架，消除传统欧拉-马鲁尼亚时间离散带来的偏差
- 避免显式计算海森矩阵，保持计算效率，适用于高维问题
- 适合需要高精度与高效性结合的科学计算场景
基于后向随机微分方程（BSDE）的深度学习方法已成为求解高维偏微分方程（PDEs）的有力替代方案，相较于物理信息神经网络（PINNs），能有效缓解维度诅咒。通过概率表示，BSDE方法通常可实现无需二阶导数的训练目标。然而，近期研究发现，常用的欧拉-马鲁尼亚（EM）时间离散化会引入内在偏差。尽管高阶方法如海恩（Heun）可完全消除该偏差，但会重新引入空间二阶导数并带来显著计算开销。本文系统分析了EM导致的损失偏差，并提出一种无偏、无需二阶导数的训练框架，保留了BSDE方法的计算优势。代码已开源：https://github.com/seojaemin22/Un-EM-BSDE。
原文摘要 · Abstract (English)
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.
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