Numerical Methods of Neural Network Discretization for Solving Nonlinear Differential Equations

Data Science Seminar

Wenrui Hao (The Pennsylvania State University)

Abstract

Machine learning-based approaches for solving differential equations have gained widespread attention, with neural network-based discretization emerging as a powerful technique. This approach, by parameterizing a set of neural network functions, offers a compelling method for solving differential equations. Various methodologies, including the deep Ritz method and physics-informed neural networks, have been developed to compute numerical solutions. To address the optimization challenges, a range of training algorithms, such as gradient descent and greedy algorithms, have been proposed. In this talk, I will introduce the Gauss-Newton method for computing numerical solutions and provide the superlinear convergence analysis on the semi-regular zeros of the vanishing gradient. Additionally, I will present a novel approach based on Homotopy Physics-Informed Neural Networks (HomPINNs), which integrates Physics-Informed Neural Networks (PINNs) with the homotopy continuation method. This innovative framework efficiently solves nonlinear elliptic differential equations with multiple solutions. Our results highlight how machine learning approaches handle differential equations with multiple solutions and unknown parameters.