Scale-Decomposed Physics-Informed Neural Networks for Singular Perturbation Problems

A novel physics-informed neural network (PINN), scale-decomposed PINN (SD-PINN), is proposed for singular perturbation problems, such as boundary-layer flows. Singular perturbation problems exhibit sharp changes in solutions at different scales, owing to large gradients in equations, which presents a challenge to a conventional PINN. In SD-PINN, the solutions at different scales are represented by different neural networks, and their matching is achieved by imposed matching conditions. In comparison with the conventional procedure for singular perturbation problems, no domain decomposition is used. A nonlinear stretching transformation is introduced to prevent the occurrence of semi-infinite regions in the neural networks. Six benchmark singular perturbation problems are used to evaluate SD-PINN, including linear and nonlinear second-order ordinary differential equations, two-dimensional hyperbolic problems, and a flat-plate boundary-layer flow. It is demonstrated that SD-PINN can reproduce asymptotic and non-asymptotic solutions for small and finite perturbation parameters, respectively. Application of SD-PINN to the inference of perturbation parameters from available data is also discussed. Codes are available at https://github.com/LeiZhang-code/SD-PINN.