A pseudo-time stepping and parameterized physics-informed neural network framework for Navier–Stokes equations

Physics-informed neural networks (PINNs) have emerged as a popular approach in scientific machine learning for solving both forward and inverse problems of partial differential equations (PDEs). However, complex physical systems are often characterized by parameters, such as viscosity and Reynolds number in fluid dynamics, which pose significant challenges for parameterized PDE solutions. The inherent limitations of PINNs include the need for repeated and time-consuming training under varying parameter conditions, and the minimization of PDE residuals with PDE-based soft constraints, which makes the “ill-conditioned” problem. To address these issues, this paper proposes an innovative framework: the pseudo-time stepping and parameterized physics-informed neural network (P2PINN). P2PINN leverages the explicit encoding of only two PDE parameters' latent representations to enable efficient interpolation and extrapolation across a wide range of parameters. By integrating the pseudo-time stepping method with deep learning, the framework significantly alleviates the ill-conditioned problem. We validated our method in the context of the Navier–Stokes equations, and experimental results demonstrate that P2PINN achieves solution speedups of up to 2–4 orders of magnitude compared to baseline PINNs and their variants, while also surpassing them in accuracy.