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arXiv 2603.15331math.NAcs.LGcs.NA
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一种扩展的TW-PINN：用于具有通用系数的反应扩散方程行波解的物理信息神经网络

A scaled TW-PINN: A physics-informed neural network for traveling wave solutions of reaction-diffusion equations with general coefficients

Department of Mathematics \& POSTECH MINDS (Mathematical Institute for Data Science), Pohang University of Science；Graduate School of Artificial Intelligence \& POSTECH MINDS (Mathematical Institute for Data Science), Pohang University of Science

Seungwan Han, Kwanghyuk Park, Jiaxi Gu, Jae-Hun Jung

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AI总结：

本文提出了一种高效的物理信息神经网络框架，用于计算具有不同反应和扩散系数的n维反应扩散方程的行波解。通过应用行波形式的缩放变换，将原问题转化为具有单位反应和扩散系数的一维缩放反应扩散方程。该方法在不同系数选择和空间维度下重用单一PINN求解器，并证明了其对行波解的通用逼近性质。

摘要：

我们提出了一种高效且通用的物理信息神经网络（PINN）框架，用于计算具有各种反应和扩散系数的n维反应扩散方程的行波解。通过应用行波形式的缩放变换，将原问题转化为具有单位反应和扩散系数的一维缩放反应扩散方程。这种转换导致了所提出的框架，称为缩放TW-PINN，其中单一PINN求解器在缩放方程上训练后，可重用于不同的系数选择和空间维度。我们还证明了所提出PINN求解器对行波解的通用逼近性质。一维和二维的数值实验，以及与现有wave-PINN方法的比较，展示了缩放TW-PINN的准确性、灵活性和优越性能。最后，我们探索了该框架扩展到具有通用初始条件的Fisher方程的应用。

英文摘要：

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.

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