Novel analysis of physics-informed neural networks for solving Cahn-Allen, FitzHugh-Nagumo and Fisher–KPP Models

Abstract

This paper presents a hybrid semi-analytical/deep-learning framework for nonlinear PDEs, combining the New Iterative Method (NIM) with Physics-Informed Neural Networks (PINNs). A truncated NIM series provides a closed-form baseline satisfying the initial condition exactly, while a neural network learns only the residual correction – shown to be one to two orders of magnitude smaller in amplitude than the full solution, which structurally eases the network’s learning task. We evaluate the approach on three nonlinear reaction–diffusion equations: Cahn–Allen equation and FitzHugh–Nagumo equation (cubic reaction terms) and Fisher–KPP equation (quadratic term). Against a matched standard PINN, the hybrid model reduces mean absolute error by roughly an order of magnitude on FitzHugh–Nagumo, by a smaller but consistent margin on Fisher–KPP, and is modestly outperformed by the standard PINN on Cahn–Allen; these results hold across four independent seeds per method, and both PINN-based methods outperform plain NIM truncation on every equation (by threefold to nearly twentyfold). We further validate against an independent finite-difference solver, a depth/width sensitivity study, and a multi-seed noise-robustness study under Gaussian initial-condition perturbation, all at full training scale for all three equations. A convergence analysis for both the NIM series and the hybrid scheme is given, with explicit Lipschitz and local-uniqueness hypotheses, separating analytical truncation error from optimisation error. Together, the results show that coupling a semi-analytical baseline with a physics-informed neural correction is effective for nonlinear reaction–diffusion problems, and precisely characterise where its advantage over a standard PINN is largest, where it disappears, and where it reverses.