Approximating Families of Sharp Solutions to Fisher's Equation with Physics-Informed Neural Networks

TL;DR

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.

Abstract

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.

Figures (6)

• Figure 1: Schematic drawing of the wave-PINN that either approximates $u(x,t)$, or $u(x,t;\rho)$ by taking $\rho$ as an additional input variable. The additional wave layer (see Fig. \ref{['fig:wave_layer']}) constrains the network function to $u_\theta(\tilde{z})$, thus ensuring the form of traveling wave solutions. The standard architecture does not use the wave layer. The generalizing architecture uses $\rho$ as an additional input.
• Figure 2: Schematic structure of the wave layer, which compresses the input variables into the form $\tilde{z}(x,t) = \theta_1 x + \theta_2 t + \theta_3$ for the single-$\rho$ approximation, or $\tilde{z}(x,t;\rho) = \theta_1 \rho_1 x + \theta_2 \rho_2 t + \theta_3$ when using the generalizing architecture. Specific transformations in the feature scaling step are used to obtain $\rho_1$ and $\rho_2$, and to ease the adjustment of $\theta_1$, $\theta_2$, and $\theta_3$.
• Figure 3: Wave front for $\rho=10^4$ and $t=0.002$. Left: Reference solution and prediction of the wave-PINN with the fully trained model using $\lambda=1$. Right: Weighted physics residuals for different values of $\lambda$ (see Eq. \ref{['eq:collocation_weighting']}), assessed for the predicted wave front in the left plot. Unweighted physics residuals ($\lambda=0$) exhibit orders of magnitude larger values at transition points on the wave front compared to those outside the wave front.
• Figure 4: Learning curves for the four tested models for $\rho=10^3$, using ten uniquely initialized instances per model. For the PINN models, results are shown for $\lambda=1$ only. (Top row) Training set errors, evaluated using the total loss function in Eq. \ref{['eq:total_loss_fisher']} (Bottom row) Test set errors, evaluated by calculating the data loss from Eq.\ref{['eq:data_loss_function']} on randomly sampled test points within the computational domain.
• Figure 5: The median (and 25%- and 75%-quantiles) of the $L_2$-error for each tested model on the continuous domain $\rho\in[100, 10.000]$. Labeled training data was used at $\rho=10^2$ and $\rho=10^4$, indicated by the dashed black lines.