Error Analysis of Physics-Informed Neural Networks for Approximating Dynamic PDEs of Second Order in Time
This work addresses the theoretical underpinnings of PINNs for dynamic PDEs, offering incremental improvements in error estimation and algorithm design for researchers in computational physics and machine learning.
The paper tackles the error analysis of physics-informed neural networks (PINNs) for approximating dynamic second-order PDEs, providing bounds on approximation errors based on training loss and data points, and introduces a variant PINN algorithm with new loss forms that improve performance in numerical experiments.
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.