Physics-Informed Neural Networks for Solving Differential Equations

Abstract

Differential equations (DEs) are fundamental tools for modeling physical phenomena across various scientific and engineering disciplines. Traditional numerical methods for solving these equations often require extensive computational resources, especially when dealing with high-dimensional, nonlinear, or data-scarce problems. In recent years, Physics-Informed Neural Networks (PINNs) have emerged as a powerful alternative, blending the strengths of deep learning with the rigor of physical laws. By embedding DEs, initial conditions, and boundary conditions directly into the loss function of a neural network, PINNs enable the solution of both forward and inverse problems without the need for mesh generation or large datasets. This paper presents an overview of the PINN methodology and applies it to solve a one-dimensional heat conduction problem without relying on empirical data. The results demonstrate that PINNs can accurately approximate the analytical solution, confirming their potential as flexible, mesh-free solvers. The advantages and challenges of PINNs are also discussed, highlighting their role in advancing data-driven scientific computing.