Physics Informed by Deep Learning: Numerical Solutions of Modified Korteweg-de Vries Equation

Abstract

In this paper, with the aid of symbolic computation system Python and based on the deep neural network (DNN), automatic differentiation (AD), and limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) optimization algorithms, we discussed the modified Korteweg-de Vries (mkdv) equation to obtain numerical solutions. From the predicted solution and the expected solution, the resulting prediction error reaches $10^{-6}$ . The method that we used in this paper had demonstrated the powerful mathematical and physical ability of deep learning to flexibly simulate the physical dynamic state represented by differential equations and also opens the way for us to understand more physical phenomena later.