Numerical methods for solving nonlinear PDEs from homotopy methods to machine learning

Applied and Computational Mathematics Seminar
Monday, October 12, 2020 - 2:00pm for 1 hour (actually 50 minutes)
Wenrui Hao – Penn State University – wxh64@psu.edu
Yingjie Liu

Many systems of nonlinear PDEs are arising from engineering and biology and have attracted research scientists to study the multiple solution structure such as pattern formation. In this talk, I will present several methods to compute the multiple solutions of nonlinear PDEs. In specific, I will introduce the homotopy continuation technique to compute the multiple steady states of nonlinear differential equations and also to explore the relationship between the number of steady-states and parameters. Then I will also introduce a randomized Newton's method to solve the nonlinear system arising from neural network discretization of the nonlinear PDEs. Several benchmark problems will be used to illustrate these ideas.