APPLICATIONS OF MACHINE LEARNING IN SOLVING DIFFERENTIAL EQUATIONS

Abstract

Differential equations constitute a fundamental tool for modeling dynamical system in mathematics. Classical numerical schemes such as finite difference, finite element, and spectral methods provide well-established approaches for approximating solutions but often rely on mesh generation, fine discretization, and problem-specific formulations. In recent years, machine learning (ML) has emerged as a flexible alternative framework in which the solution of a differential equation is approximated by a parameterized function, typically a neural network, trained to satisfy the governing equations and associated boundary or initial conditions. In this work, we investigate the capacity of ML-based solvers to approximate solutions of ordinary and partial differential equations. The methodology exploits universal approximation theorems and automatic differentiation to obtain not only the solution but also its derivatives, enabling direct enforcement of differential operators. Numerical illustrations include oscillatory ordinary differential equations, the nonlinear viscous Burgers’ equation, and the two-dimensional heat equation. The results demonstrate that ML solvers achieve accurate approximations of solutions and their derivatives, reproduce qualitative features such as phase-space invariants and shock profiles, and exhibit systematic convergence with increasing collocation points. The findings suggest that ML provides a mathematically consistent and mesh-free alternative to classical methods, while raising open questions regarding computational efficiency, error analysis, and rigorous convergence guarantees.