General-Kindred Physics-Informed Neural Network to the Solutions of Singularly Perturbed Differential Equations

Abstract: Physics-Informed Neural Networks (PINNs) have become a promising research direction in the field of solving Partial Differential Equations (PDEs). Dealing with singular perturbation problems continues to be a difficult challenge in the field of PINN. The solution of singular perturbation problems often exhibits sharp boundary layers and steep gradients, and traditional PINN cannot achieve approximation of boundary layers. In this manuscript, we propose the General-Kindred Physics-Informed Neural Network (GKPINN) for solving Singular Perturbation Differential Equations (SPDEs). This approach utilizes asymptotic analysis to acquire prior knowledge of the boundary layer from the equation and establishes a novel network to assist PINN in approximating the boundary layer. It is compared with traditional PINN by solving examples of one-dimensional, two-dimensional, and time-varying SPDE equations. The research findings underscore the exceptional performance of our novel approach, GKPINN, which delivers a remarkable enhancement in reducing the $L_2$ error by two to four orders of magnitude compared to the established PINN methodology. This significant improvement is accompanied by a substantial acceleration in convergence rates, without compromising the high precision that is critical for our applications. Furthermore, GKPINN still performs well in extreme cases with perturbation parameters of ${1\times10}^{-38}$, demonstrating its excellent generalization ability.