This paper employs physics-informed neural networks (PINNs) to solve Fisher's equation, a fundamental reaction-diffusion system with both simplicity and significance. The focus is on investigating Fisher's equation under conditions of large reaction rate coefficients, where solutions exhibit steep traveling waves that often present challenges for traditional numerical methods. To address these challenges, a residual weighting scheme is introduced in the network training to mitigate the difficulties associated with standard PINN approaches. Additionally, a specialized network architecture designed to capture traveling wave solutions is explored. The paper also assesses the ability of PINNs to approximate a family of solutions by generalizing across multiple reaction rate coefficients. The proposed method demonstrates high effectiveness in solving Fisher's equation with large reaction rate coefficients and shows promise for meshfree solutions of generalized reaction-diffusion systems.

翻译：本文采用物理信息神经网络（PINNs）求解Fisher方程——一个兼具简洁性与重要性的基本反应-扩散系统。研究重点聚焦于大反应速率系数条件下Fisher方程的求解问题，该条件下的解呈现陡峭行波形态，常对传统数值方法构成挑战。为应对这些挑战，本文在网络训练中引入残差加权方案，以缓解标准PINN方法面临的困难。同时，研究还探索了专门用于捕捉行波解的网络架构设计。本文通过跨多个反应速率系数的泛化能力评估了PINNs逼近解族的效果。所提方法在求解大反应速率系数的Fisher方程中表现出高效性，并为广义反应-扩散系统的无网格求解提供了可行路径。