Methods to Improve the Training of Physics-Informed Neural Networks

Physics-Informed Neural Networks (PINNs) are a tool used for finding numerical solutions to partial differential equations. PINNs can be difficult to train because the optimization problem that arises is non-convex. So, even if a minimum is found, it is difficult to know if this is a global or just local minimum. We propose methods to improve their training, including a multilevel training scheme. Next, we relate the first-order formulation of PINNs to the least-square finite element method. Then we compare the performance of the first and second-order formulations for Laplace’s equation with a discontinuous solution and a simple nonlinear PDE. We consider the cases where interior data is and is not available as well as the impact of noisy data on the results.

Last modified: Wed Mar 27 13:25:49 EDT 2024