Neural networks are universal approximators and are studied for their use in solving differential equations. However, a major criticism is the lack of error bounds for obtained solutions. This paper proposes a technique to rigorously evaluate the error bound of Physics-Informed Neural Networks (PINNs) on most linear ordinary differential equations (ODEs), certain nonlinear ODEs, and first-order linear partial differential equations (PDEs). The error bound is based purely on equation structure and residual information and does not depend on assumptions of how well the networks are trained. We propose algorithms that bound the error efficiently. Some proposed algorithms provide tighter bounds than others at the cost of longer run time.

翻译：神经网络作为通用逼近器，已被研究用于求解微分方程。然而，其主要批评在于所获解缺乏误差界。本文提出一种严格评估物理信息神经网络（PINNs）在大多数线性常微分方程（ODE）、某些非线性ODE以及一阶线性偏微分方程（PDE）上误差界的技术。该误差界完全基于方程结构与残差信息，不依赖于网络训练质量的假设。我们提出能高效界定误差的算法。部分算法以更长的运行时间为代价，提供更紧的误差界。