A scaled TW-PINN: A physics-informed neural network for traveling wave solutions of reaction-diffusion equations with general coefficients
This work addresses a computational challenge in physics-informed neural networks for researchers in applied mathematics and computational science, though it appears incremental as it builds on existing PINN methods.
The authors tackled the problem of computing traveling wave solutions for reaction-diffusion equations with varying coefficients by proposing a scaled TW-PINN framework, which reduces the problem to a one-dimensional scaled equation and demonstrates accuracy and superior performance in numerical experiments.
We propose an efficient and generalizable physics-informed neural network (PINN) framework for computing traveling wave solutions of $n$-dimensional reaction-diffusion equations with various reaction and diffusion coefficients. By applying a scaling transformation with the traveling wave form, the original problem is reduced to a one-dimensional scaled reaction-diffusion equation with unit reaction and diffusion coefficients. This reduction leads to the proposed framework, termed scaled TW-PINN, in which a single PINN solver trained on the scaled equation is reused for different coefficient choices and spatial dimensions. We also prove a universal approximation property of the proposed PINN solver for traveling wave solutions. Numerical experiments in one and two dimensions, together with a comparison to the existing wave-PINN method, demonstrate the accuracy, flexibility, and superior performance of scaled TW-PINN. Finally, we explore an extension of the framework to the Fisher's equation with general initial conditions.