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P3: Exploration on Applications of Machine Learning Methods in Approximating Parameter-Dependent Partial Differential Equations
Author: Jiajing Guan , Advisor: Howard Elman (CS Department)
Problem Statement Presentation
Abstract
In this report, we examined the performance of machine learning algorithms on approximating solutions of parameter-dependent partial differential equations. We investigated two algorithms: Proper Orthogonal Decomposition Neural Network Reduced Basis method (POD-NN RB) and Physics-Informed Neural Networks (PINN). We tested the effects of network depth, network structure and number of training samples on the accuracy of approximations produced by POD-NN RB, for an unsteady Burger’s equation and a nonlinear diffusion equation. We then found the inherent inability of PINN in approximating singularly perturbed problems, such as convection-diffusion equations. We utilized techniques used in singular perturbation theory to improve the accuracy of approximations produced by PINN drastically.
Final Presentation ,
Final Report
