为物理信息神经网络求解微分方程提供精确与近似误差界
Exact and approximate error bounds for physics-informed neural networks
- 基于残差和方程结构推导通用误差表达式
- 提出通用情形近似界与特定情形精确界计算方法
- 无需依赖数值解即可获得可靠误差估计，适合科学计算可信度评估
神经网络求解微分方程作为传统数值方法的替代方案近年来受到关注。然而，现有误差界仅适用于特定方程。本文在非线性一阶常微分方程的物理信息神经网络（PINN）解上取得重要进展，给出了一般形式的解误差表达式。进一步提出一种用于一般情况的近似误差界计算方法，以及特定情况下的精确误差界计算方法。所有误差界仅依赖于残差信息与方程结构，无需借助数值解。在若干具体案例中验证了该方法的有效性，证明其可在不依赖真实解的情况下成功提供误差界。
原文摘要 · Abstract (English)
The use of neural networks to solve differential equations, as an alternative to traditional numerical solvers, has increased recently. However, error bounds for the obtained solutions have only been developed for certain equations. In this work, we report important progress in calculating error bounds of physics-informed neural networks (PINNs) solutions of nonlinear first-order ODEs. We give a general expression that describes the error of the solution that the PINN-based method provides for a nonlinear first-order ODE. In addition, we propose a technique to calculate an approximate bound for the general case and an exact bound for a particular case. The error bounds are computed using only the residual information and the equation structure. We apply the proposed methods to particular cases and show that they can successfully provide error bounds without relying on the numerical solution.
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