Deep Learning-Based PDE Solver: PINN Versus Classical Method for the 1D Heat Equation

PINN Versus Classical Method for the 1D Heat Equation

Abstract

Partial Differential Equations (PDEs) play a vital role in modeling heat transfer and diffusion phenomena in science and engineering, ensuring the development of accurate and efficient numerical solvers for consistent simulations. Conventional discretization-based procedures often involve mesh production for regular geometries and frequently encounter complications for irregular cases and sparse data. Currently, Machine Learning approaches, specifically Physics-Informed Neural Networks (PINNs), have emerged as capable mesh-free alternatives that integrate governing physical rules directly into the learning method. In this paper, PINNs have been applied to a One-Dimensional (1D) heat equation. The PINN model has been formulated by using DeepXDE and compared with the Finite Difference Method (FDM). The governing equation (GE), along with initial and boundary conditions, is rooted in the loss function, which monitors the training process, allowing the PINN to achieve physical stability by learning the result over the entire spatio-temporal field. The application influences programmed differentiation for the resulting derivatives and reduces the residual error, rather than relying on explicit discretization. Results suggest that the neural networks successfully approximate the solution to the heat equation with competing error rates. Moreover, it is flexible for noisy data and complex domains. Comparative convergence behavior and visualization results are presented to demonstrate the effectiveness of the PINN framework.