Comparison and Analysis of Neural Solver Methods for Differential Equations in Physical Systems

Abstract

Differential equations are ubiquitous in many fields of study, yet not all equations, whether ordinary or partial, can be solved analytically. Traditional numerical methods such as time-stepping schemes have been devised to approximate these solutions. With the advent of modern deep learning, neural networks have become a viable alternative to traditional numerical methods. By reformulating the problem as an optimisation task, neural networks can be trained in a semi-supervised learning fashion to approximate nonlinear solutions. In this paper, neural solvers are implemented in TensorFlow for a variety of differential equations, namely: linear and nonlinear ordinary differential equations of the first and second order; Poisson’s equation, the heat equation, and the inviscid Burgers’ equation. Different methods, such as the naive and ansatz formulations, are contrasted, and their overall performance is analysed. Experimental data is also used to validate the neural solutions on test cases, specifically: the spring-mass system and Gauss’s law for electric fields. The errors of the neural solvers against exact solutions are investigated and found to surpass traditional schemes in certain cases. Although neural solvers will not replace the computational speed offered by traditional schemes in the near future, they remain a feasible, easy-to-implement substitute when all else fails.

Keywords

• differential equations, deep learning, neural networks, numerical methods.