Machine Learning methods for numerical solutions to PDE

Physics-Informed Neural Networks (PINNs) have achieved significant success as a machine learning method (using artificial neural networks) for numerically solving differential equations. We use the Deep Ritz method to solve partial differential equations (PDEs) in variational form, including boundary-value problems and eigenvalue problems of Laplace equations and obstacle problems. 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. Further applications of PINN to solve fractional differential equations will also be illustrated.

The link to the slides is given below:
https://mathpost.asu.edu/~gyu/