We consider the approximation of a class of dynamic partial differential equations (PDE) of second order in time by the physics-informed neural network (PINN) approach, and provide an error analysis of PINN for the wave equation, the Sine-Gordon equation and the linear elastodynamic equation. Our analyses show that, with feed-forward neural networks having two hidden layers and the $\tanh$ activation function, the PINN approximation errors for the solution field, its time derivative and its gradient field can be effectively bounded by the training loss and the number of training data points (quadrature points). Our analyses further suggest new forms for the training loss function, which contain certain residuals that are crucial to the error estimate but would be absent from the canonical PINN loss formulation. Adopting these new forms for the loss function leads to a variant PINN algorithm. We present ample numerical experiments with the new PINN algorithm for the wave equation, the Sine-Gordon equation and the linear elastodynamic equation, which show that the method can capture the solution well.

翻译：我们考虑使用物理信息神经网络（PINN）方法逼近一类时间二阶动态偏微分方程（PDE），并针对波动方程、Sine-Gordon方程和线性弹性动力学方程提供PINN的误差分析。我们的分析表明，使用具有两个隐藏层和tanh激活函数的前馈神经网络时，PINN对解场、其时间导数及其梯度场的逼近误差可以通过训练损失和训练数据点（求积点）数量得到有效控制。为进一步，我们的分析提出了训练损失函数的新形式，其中包含对误差估计至关重要但在标准PINN损失公式中缺失的特定残差项。采用这些新形式的损失函数将引出一种变体PINN算法。我们针对波动方程、Sine-Gordon方程和线性弹性动力学方程进行了大量数值实验，结果表明该方法能够良好地捕获解。