NeuroDiffEq: A Python package for solving differential equations with neural networks

Abstract

Differential equations emerge in various scientific and engineering domains for modeling physical phenomena. Most differential equations of practical interest are analytically intractable. Traditionally, differential equations are solved by numerical methods. Sophisticated algorithms exist to integrate differential equations in time and space. Time integration techniques continue to be an active area of research and include backward difference formulas and Runge-Kutta methods Common spatial discretization approaches include the finite difference method (FDM), finite volume method (FVM), and finite element method (FEM) as well as spectral methods such as the Fourier-spectral method. These classical methods have been studied in detail and much is known about their convergence properties. Moreover, highly optimized codes exist for solving differential equations of practical interest with these techniques While these methods are efficient and well-studied, their expressibility is limited by their function representation.