Abstract

(joint work with Javier Omella, Jon A. Rivera, and Jamie M. Taylor) The Ritz method is a traditional method for solving symmetric and positive definite problems governed by Partial Differential Equations (PDEs). This method minimizes the energy functional, and a first Neural Network (NN) formulation using this method was proposed in [1]. In this talk, we first illustrate how traditional methods for solving PDEs using NNs (like Deep-Ritz, Deep Least-Squares, and other Deep Galerkin methods) may suffer from strong quadrature problems, leading to poor approximate solutions. We envision four alternatives to overcome this challenge: a) Monte Carlo methods, b) adaptive integration, c) piecewise-polynomial approximations of the NN solution, and d) the inclusion of regularization terms in the loss following the ideas of [2]. From all these methods, we develop an r-adaptive method, which falls under the category of piecewise-polynomials approximations of the NN. We consider a piecewise-linear solution over a grid—allowing for exact integration—and simultaneously optimize the node positions (r-adaptivity) and the solution values. We show promising numerical results of the r-adaptive Deep Ritz method in one- and two-dimensional domains.

Weinan E and Bing Yu, The Deep Ritz Method: A Deep Learning-Based Numerical Algorithm for Solving Variational Problems. Commun. Math. Stat., vol. 6, no. 1, pp. 1–12 (2018). https://doi.org/10.1007/s40304-018-0127-z

Siddhartha Mishra and Roberto Molinaro, Estimates on the generalization error of physics informed neural networks (PINNs) for approximating PDEs. arXiv preprint arXiv:2006.16144  (2020).