Physics Informed Neural Networks(PINNs) Approach for Numerical Solution of Heat Equation

Abstract

Heat conduction is a fundamental physical process governing thermal energy transport across diverse media. While traditional numerical methods (e.g., Finite Difference) are physically powerful, they often require complex and can be computationally expensive for high dimension. PINNs utilize the universal approximation power of neural networks constrained by the underlying physical laws. We present a Python-based implementation demonstrating how PINNs can be used to visualize thermal diffusion. A physics-informed neural networks is formulated by embedding the governing heat equation, along with the prescribed initial and boundary conditions, into the loss function of a neural network. Automatic differentiation is employed to compute the required spatial and temporal derivatives. The network is trained using collocation points sampled across the space–time domain, without the need for labeled training data. The PINNs-based solution exhibits excellent agreement with the actual solution of the heat equation, correctly reproducing the spatial temperature profiles and their temporal evolution. The study highlights the effectiveness of PINNs as a reliable alternative to traditional numerical methods.