Solving the FitzHugh-Nagumo Equation Using Physics-Informed Neural Networks (PINNs) and Deep Galerkin Methods

Abstract

When the coefficients in the partial differential equations (PDEs) vary with spatial position, as in the FitzHugh-Nagumo (FHN) equations, the numerical solution is more difficult. In this work, a novel framework that combines Physics-Informed Neural Networks (PINNs) and Deep Galerkin Methods (DGM) is proposed to overcome these difficulties, which marks a paradigm shift in solving PDEs using AI. The neural network structure is the same in both methods, with hidden layers, and is over-trained using the Adam optimization algorithm. The optimization of collocation point selection based on Sobol sequences is an important component that is helpful for PDEs with sharp gradients and assures optimal distribution of collocation points. We have shown that both techniques are highly accurate in our comparative analysis for a variety of parameter regimes. This shows the complementary strengths of DGM, as it has reduced training times and, for some configurations, PINNs performs better, indicating a computational advantage of DGM. Overall, the framework demonstrates the potential of deep learning for solving complex PDEs involving variable coefficients and provides flexible and scalable solutions that can effectively capture complex boundary conditions and geometries. The results have implications for new research areas in computational neuroscience, fluid mechanics, and material science. Future avenues for improving the ability to calculate the accuracy of the solution and computational efficiency could be adaptive sampling and multi-fidelity modeling.