Embedding physical laws into Deep Neural Networks for solving generalized Burgers–Huxley equation

Abstract

Among the difficult problems in mathematics is the problem of solving partial differential equations (PDEs). To date, there is no technique or method capable of solving all PDEs despite the large number of effective methods proposed. One finds in the literature, numerical methods such as the methods of finite differences, finite elements, finite volumes and their variants, semi-analytical methods such as the Variational Iterative Method, New Iterative Method and others. In recent years, we have witnessed the introduction of neural networks in solving PDEs. In this work, we will propose an adaptation of the method of embedding some physical laws into neural networks for solving Burgers–Huxley equation and revealing the dynamic behavior of the equation directly from spatio-temporal data. We will combine our technique with the Residual-based Adaptive Refinement method to improve its accuracy. We will give a comparison of the proposed method with those obtained by the New Iterative Method.