新框架可高效求解带任意系数的反应扩散波解，通用性强且精度高。
A scaled TW-PINN: A physics-informed neural network for traveling wave solutions of reaction-diffusion equations with general coefficients
- 通过波形缩放将多维问题转为一维标准方程，统一求解
- 单个模型适配不同系数与维度，数值实验验证其高精度
- 适用于复杂初始条件，适合研究波动现象的科研人员
我们提出一种高效且可泛化的物理信息神经网络（PINN）框架，用于计算具有各类反应与扩散系数的n维反应-扩散方程的行进波解。通过施加基于行进波形式的缩放变换，原问题被简化为一个具有单位反应和扩散系数的一维缩放反应-扩散方程。由此构建的缩放行进波PINN（scaled TW-PINN）框架中，单一训练好的PINN求解器可复用于不同系数选择和空间维度。我们还证明了所提PINN求解器对行进波解具备普适逼近性。一维和二维数值实验，以及与现有wave-PINN方法的对比，展示了scaled TW-PINN在精度、灵活性和性能上的优越性。最后，我们还将该框架扩展至具一般初值的Fisher方程。
原文摘要 · Abstract (English)
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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