Solving the Fisher nonlinear differential equations via Physics-Informed Neural Networks: A Comprehensive Retraining Study and Comparative Analysis with the Finite Difference Method

TL;DR

This paper demonstrates that Physics-Informed Neural Networks can accurately solve the nonlinear Fisher-KPP equation, comparing their performance with finite difference methods, and explores retraining strategies to optimize model accuracy and efficiency.

Contribution

It provides a comprehensive methodology for applying PINNs to nonlinear PDEs, including retraining strategies and a detailed comparison with traditional numerical methods.

Findings

PINNs accurately approximate the Fisher-KPP equation solutions.

Retraining strategies significantly improve PINN performance.

PINNs are competitive with finite difference methods for nonlinear PDEs.

Abstract

Physics-Informed Neural Networks (PINNs) represent a groundbreaking paradigm in scientific computing, seamlessly integrating the robust framework of deep learning with fundamental physical laws. This paper meticulously applies the standard PINN framework to solve the challenging one-dimensional nonlinear Fisher-KPP equation, a critical model in reaction-diffusion dynamics describing phenomena such as population spread and flame propagation. We detail a comprehensive methodology, encompassing the neural network architecture, the physics-informed loss function, and an in-depth investigation into retraining strategies aimed at optimizing model performance. Our approach is rigorously validated through a direct comparison of the PINN solution against both the known analytical solution and a numerical solution derived from the Finite Difference Method (FDM). Through this work, we elucidate the intricate balance between model complexity, training efficiency, and accuracy. Results highlight the PINN's remarkable capability in accurately approximating the solution to this complex PDE, while also shedding light on the critical aspects and challenges of model retraining, particularly concerning the optimizer's state. This study provides a thorough quantitative error analysis, demonstrating the efficacy of PINNs as a viable and competitive alternative to traditional numerical methods for solving nonlinear differential equations, and discusses their broader applications across various scientific domains.