Asymptotic Physics-Informed Neural Networks for Solving Singularly Perturbed Problems

Recently, physics-informed neural networks (PINNs) have made great progress in scientific machine learning, especially in clarifying physical systems and phenomena defined by partial differential equations (PDEs). However, PINNs fail to solve PDEs with special properties, such as singularly perturbed differential equations (SPDEs). SPDEs tend to have boundary layers, where the value of the solution increases or decreases drastically. To address this issue, the method called asymptotic PINNs (A-PINNs) is proposed, which combines the prior knowledge provided by the Shishkin mesh with domain decomposition methods to solve SPDEs effectively. Numerical results indicate that our method shows superiority in handling the singularly perturbed property of SPDEs.