Singular layer PINN methods for steep reaction-diffusion equations in a smooth convex domain

Abstract

We introduce a novel semi-analytic method for solving singularly perturbed reaction-diffusion problems in a smooth domain using neural network architectures. To manage steep solution transitions near the boundary, we utilize the boundary-fitted coordinates and perform boundary layer analysis to construct a corrector function which describes the singular behavior of the solution near the boundary. By integrating the boundary layer corrector into the conventional PINN structure, we propose our new sl-PINNs (singular-layer Physics-Informed Neural Networks). The sl-PINN framework is specifically designed to capture sharp transitions inside boundary layers, significantly improving the approximation accuracy for solutions under small perturbation parameters. The computational results of various simulations in this article demonstrate the superior performance of sl-PINNs over conventional PINNs in handling such problems.