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 physical laws directly into neural-network training. Recent studies have shown that parameterized PINNs can learn families of Navier–Stokes solutions across Reynolds numbers by treating governing parameters as additional network inputs. However, the accuracy of data-free PINNs deteriorates significantly in convection-dominated high-Reynolds-number flows because of optimization stiffness and sharp multiscale flow structures. Motivated by these limitations, the present study investigates a sparse-supervised hybrid parameterized PINN framework for incompressible Navier–Stokes flows, emphasizing regime-aware learning, localized Reynolds-number supervision, and data-efficient correction of high-Reynolds-number failure modes. The Reynolds number is incorporated explicitly as a network input, enabling a single model to learn a continuous parametric flow manifold across different operating conditions. The methodology is first demonstrated for two-dimensional lid-driven cavity flow and further assessed using backward-facing step flow to evaluate generalizability to a qualitatively different separated-flow configuration. For low Reynolds numbers, data-free PINNs accurately recover velocity and pressure fields using only governing equations and boundary conditions. At higher Reynolds numbers, where convection-dominated effects degrade physics-only PINNs, a hybrid framework combining transfer learning and localized sparse computational fluid dynamics (CFD) supervision is introduced. Although the training range spans 500<Re<1000, supervised CFD data are provided only within the restricted interval 750<Re<850. Supervision is additionally limited to only 3%−20% of the computational points to quantify minimum data requirements. The results demonstrate that approximately 5% supervised data are sufficient to recover accurate solutions with high predictive fidelity. Comparisons with CFD simulations demonstrate strong agreement in velocity, pressure, vorticity, and reattachment characteristics across interpolation and limited extrapolation regimes. The study highlights a practical physics-first hybridization strategy in which sparse data are introduced only when data-free PINNs become insufficient, providing a data-efficient framework for incompressible flows across Reynolds numbers.