Approximating Families of Sharp Solutions to Fisher's Equation with
  Physics-Informed Neural Networks
Franz M. Rohrhofer Stefan Posch Clemens Gößnitzer Bernhard C. Geiger
This paper employs physics-informed neural networks (PINNs) to solve Fisher's
equation, a fundamental representation of a reaction-diffusion system with both
simplicity and significance. The focus lies specifically in investigating
Fisher's equation under conditions of large reaction rate coefficients, wherein
solutions manifest as traveling waves, posing a challenge for numerical methods
due to the occurring steepness of the wave front. To address optimization
challenges associated with the standard PINN approach, a residual weighting
scheme is introduced. This scheme is designed to enhance the tracking of
propagating wave fronts by considering the reaction term in the
reaction-diffusion equation. Furthermore, a specific network architecture is
studied which is tailored for solutions in the form of traveling waves. Lastly,
the capacity of PINNs to approximate an entire family of solutions is assessed
by incorporating the reaction rate coefficient as an additional input to the
network architecture. This modification enables the approximation of the
solution across a broad and continuous range of reaction rate coefficients,
thus solving a class of reaction-diffusion systems using a single PINN
instance.