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.

通过隐藏层级连接的物理信息神经网络方法，结合了隐藏层级连接的前馈神经网络、改进的时间步进策略和逼近偏微分方程的物理信息方法。我们分析了该方法在两种类型的偏微分方程（抛物型和双曲型）中的收敛性和误差限定，并展示了通过长时间段的动态模拟来有效控制其解的逼近误差。该方法原则上允许两个或更多个隐藏层级，并且在第二个隐藏层级之后，可以使用任意常用的平滑激活函数。基于所提出的算法，我们提出了适用于这些偏微分方程的适当训练损失函数，这使得我们的方法与标准的物理信息神经网络（PINN）形式有所不同。通过大量的数值实验证实了该方法的有效性，并验证了理论分析中的某些方面。