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：我们考虑采用物理知识驱动的神经网络 (PINN) 方法来近似一类二阶动态偏微分方程，并提供了对波动方程、正弦戈登方程和线性弹性动力学方程的 PINN 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，PINN 的解场、其时间导数和其梯度场的近似误差可以被训练损失和训练数据点数 (积分点) 有效地限制。我们的分析进一步提出了新的损失函数形式，其中包含了某些关键残差，这些残差对误差估计至关重要，但在经典的 PINN 损失函数公式中将被忽略。采用这些新的损失函数形式可导出 PINN 算法变体。我们对波动方程、正弦戈登方程和线性弹性动力学方程进行了大量数值实验，结果表明该方法可以很好地捕捉解的特征。