Higher-Order Physics-Informed Neural Networks for Parametric Solution of The Unsteady Heat Equation

This work presents an efficient physics-informed neural network (PINN) framework for the solution of the parametric unsteady heat equation. The proposed approach investigates shallow neural network architectures based on standard perceptron networks and compares them with higher-order neural networks (HONNs) incorporating higher-order synaptic operations. The spatial coordinate, time, and thermal diffusivity coefficient are treated as network inputs, enabling a unified parametric representation of transient heat conduction. The governing heat equation, along with the associated initial and boundary conditions, is enforced through physics-informed loss functions. Numerical results demonstrate that HONNs achieve accurate solutions of the unsteady heat equation using an order of magnitude fewer trainable parameters than conventional perceptron-based PINNs, while simultaneously providing improved solution accuracy and generalization across varying thermal diffusivity values.