PINN based on multi-scale strategy for solving Navier–Stokes equation

Neural networks combined with automatic differentiation technology provide a fundamental framework for the numerical solution of partial differential equations. This framework constitutes a loss function driven by both data and physical models, significantly enhancing generalization capabilities. Combining the framework and the idea of multi-scale methods in traditional numerical methods, such as domain decomposition and collocation self-adaption, we construct a method of the Physics-Informed Neural Networks (PINNs) based on multi-scale strategy to solve Navier–Stokes equations, and the results are more effective than XPINNs and SAPINNs. The computational efficiency of the proposed method is verified by solving two-dimensional and three-dimensional Navier–Stokes equations.