为物理信息神经网络在福克-普朗克方程中的解提供可计算的误差界
Error Bounds for Physics-Informed Neural Networks in Fokker-Planck PDEs
- 基于物理约束构建神经网络，逼近概率密度函数
- 理论推导出紧致误差界，实测验证其在高维混沌系统中有效
- 适用于需快速求解概率分布的科研与工程场景
随机微分方程常用于描述随机过程的演化，其状态不确定性最佳由概率密度函数（PDF）表示，而PDF的演化由福克-普朗克偏微分方程（FP-PDE）控制。然而，通常无法解析求解FP-PDE。本文证明，物理信息神经网络（PINNs）可用于近似求解该PDF。主要贡献在于建立了一套理论框架，以构造紧密的近似误差界；同时推导出可通过标准训练方法高效构建的实用误差界。该框架还可推广至其他线性偏微分方程的近似解。在非线性、高维及混沌系统上的实验结果验证了误差界的正确性，同时展示了PINNs在获得准确PDF解方面相比蒙特卡洛方法具有显著计算加速优势。
原文摘要 · Abstract (English)
Stochastic differential equations are commonly used to describe the evolution of stochastic processes. The state uncertainty of such processes is best represented by the probability density function (PDF), whose evolution is governed by the Fokker-Planck partial differential equation (FP-PDE). However, it is generally infeasible to solve the FP-PDE in closed form. In this work, we show that physics-informed neural networks (PINNs) can be trained to approximate the solution PDF. Our main contribution is the analysis of PINN approximation error: we develop a theoretical framework to construct tight error bounds using PINNs. In addition, we derive a practical error bound that can be efficiently constructed with standard training methods. We discuss that this error-bound framework generalizes to approximate solutions of other linear PDEs. Empirical results on nonlinear, high-dimensional, and chaotic systems validate the correctness of our error bounds while demonstrating the scalability of PINNs and their significant computational speedup in obtaining accurate PDF solutions compared to the Monte Carlo approach.
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