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 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.

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.

Figures (4)

• Figure 1: Schematic of the 7x50 Physics-Informed Neural Network (PINN) architecture. The network takes spatio-temporal coordinates $(t, x)$ as input, processes them through 7 hidden layers of 50 neurons each with Tanh activation functions, and outputs the predicted solution $u(t,x)$.
• Figure 2: Training dynamics of the I-PINN model with a 7x50 architecture. (a) The total loss function decreases steadily over 40,000 iterations, shown on a logarithmic scale. (b) The adaptive weights for the initial condition (IC) and boundary conditions (BC) quickly saturate at the predefined maximum value of $10^4$, ensuring these conditions are strictly enforced.
• Figure 3: Visual comparison of the PINN solution against the analytical solution. From left to right: the exact analytical solution, the PINN-predicted solution, and the absolute error between them over the spatio-temporal domain. The PINN solution closely mirrors the exact solution, with the largest errors localized at the steep wavefront.
• Figure 4: Comparison of the FDM solution with the exact solution at the final time $t=1.0$. Left: Exact solution profile. Center: FDM solution profile. Right: Absolute error between the FDM and exact solutions. The FDM provides a highly accurate approximation, serving as a reliable benchmark.