We present the hidden-layer concatenated physics informed neural network (HLConcPINN) method, which combines hidden-layer concatenated feed-forward neural networks, a modified block time marching strategy, and a physics informed approach for approximating partial differential equations (PDEs). We analyze the convergence properties and establish the error bounds of this method for two types of PDEs: parabolic (exemplified by the heat and Burgers' equations) and hyperbolic (exemplified by the wave and nonlinear Klein-Gordon equations). We show that its approximation error of the solution can be effectively controlled by the training loss for dynamic simulations with long time horizons. The HLConcPINN method in principle allows an arbitrary number of hidden layers not smaller than two and any of the commonly-used smooth activation functions for the hidden layers beyond the first two, with theoretical guarantees. This generalizes several recent neural-network techniques, which have theoretical guarantees but are confined to two hidden layers in the network architecture and the $\tanh$ activation function. Our theoretical analyses subsequently inform the formulation of appropriate training loss functions for these PDEs, leading to physics informed neural network (PINN) type computational algorithms that differ from the standard PINN formulation. Ample numerical experiments are presented based on the proposed algorithm to validate the effectiveness of this method and confirm aspects of the theoretical analyses.

翻译：本文提出隐层拼接物理信息神经网络（HLConcPINN）方法，该方法结合了隐层拼接前馈神经网络、改进的块时间推进策略以及物理信息方法，用于逼近偏微分方程（PDE）。我们分析了该方法对于两类偏微分方程（抛物型方程以热传导方程和Burgers方程为例，双曲型方程以波动方程和非线性Klein-Gordon方程为例）的收敛性质，并建立了其误差界。我们证明，对于长时间动态模拟，其解的逼近误差能够通过训练损失有效控制。HLConcPINN方法原则上允许使用不少于两个的任意数量隐层，且除前两层外，其余隐层可采用任意常用光滑激活函数，并具有理论保证。这推广了若干近期神经网络技术——这些技术虽具有理论保证，但在网络架构上仅限于两个隐层且使用$\tanh$激活函数。我们的理论分析进一步指导了针对这些偏微分方程的合适训练损失函数的构建，从而产生了与标准物理信息神经网络（PINN）形式不同的PINN类型计算算法。基于所提算法的大量数值实验验证了该方法的有效性，并证实了理论分析的若干方面。