Approximating Families of Sharp Solutions to Fisher's Equation with Physics-Informed Neural Networks
This work addresses numerical challenges in reaction-diffusion systems for computational physics, but it is incremental as it builds on existing PINN methods.
The paper tackled solving Fisher's equation with large reaction rate coefficients, where steep traveling waves challenge traditional methods, by using physics-informed neural networks with a residual weighting scheme and specialized architecture, achieving high effectiveness in approximating families of solutions.
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.