Physics-informed neural networks for approximating dynamic (hyperbolic) PDEs of second order in time: Error analysis and algorithms

We consider the approximation of a class of dynamic partial differential equations (PDEs) 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 nonlinear Klein-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.