Sparse-Supervised Hybrid Parameterized Physics-Informed Neural Networks for Incompressible Flows Across Reynolds Numbers

Physics-informed neural networks (PINNs) provide a mesh-free framework for solving partial differential equations by embedding governing physics into neural-network training. Recent studies have shown that parameterized PINNs can learn Navier-Stokes solutions across Reynolds numbers by treating Reynolds number as an additional network input. However, physics-only PINNs often lose accuracy in convection-dominated high-Reynolds-number flows because of optimization stiffness and multiscale flow structures. This study presents a sparse-supervised hybrid parameterized PINNs framework for incompressible Navier-Stokes flows with regime-aware learning and localized Reynolds-number supervision. The approach is demonstrated for two-dimensional lid-driven cavity flow and further validated for backward-facing step flow. At low Reynolds numbers, physics-only PINNs accurately predict velocity and pressure fields using only governing equations and boundary conditions. At higher Reynolds numbers, sparse CFD supervision combined with transfer learning is introduced to improve predictive accuracy. Although the training range spans (500 < Re < 1000), CFD supervision is applied only within (750 < Re < 850) using just (3%-20%) of computational points. Results show that approximately (5%) supervised data are sufficient for accurate flow prediction. Comparisons with CFD simulations demonstrate strong agreement in velocity, pressure, vorticity, and reattachment characteristics across interpolation and limited extrapolation regimes. The proposed framework provides a practical and data-efficient hybrid strategy for incompressible flows across varying Reynolds numbers.