  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.
