Physics-Informed Neural Networks

Physics-Informed Neural Networks (PINNs) have achieved significant success as a machine learning method (using artificial neural networks) for numerically solving differential equations. We explore the low-rank features that emerge from training PINNs to solve ordinary differential equations (ODEs) and build low-rank architectures to leverage them, achieving a reduction in model complexity. We also use the Deep Ritz method to solve partial differential equations (PDEs) in variational form, including boundary-value problems and eigenvalue problems of Laplace equations. We adopt an alternative optimizer, Ensemble Kalman Inversion (EKI), to replace stochastic gradient descent/ADAM in minimizing the proposed loss functions. This optimizer will be consistently used to train the neural networks in the aforementioned examples for solving ODEs/PDEs. Additionally, we solve inverse problems in PDEs using the PINN setup to recover unknown parameters in PDEs, given partial or sparse observations of the PDE solution.