A Parameterized Physics Informed Neural Network Solver for the Navier Stokes Equations Across Reynolds Numbers

Physics informed neural networks provide a meshfree framework for solving partial differential equations by embedding governing physical laws directly into the training process. However, most PINNs developed for fluid dynamics remain restricted to fixed flow parameters, requiring retraining for each new condition and limiting their usefulness as general purpose solvers. In this work, we develop a parameterized PINNs formulation for the incompressible Navier Stokes equations in which the Reynolds number (Re) is treated as an explicit network input, enabling a single trained model to represent a continuous family of flow solutions. The approach is demonstrated using the 2D lid driven cavity flow as a canonical benchmark. For low Re, where the flow is laminar and diffusion dominated, pure PINNs trained solely using the governing equations and boundary conditions accurately reproduce velocity and pressure fields across a wide range of Re, including cases not explicitly sampled during training. As the Re increases and the flow becomes increasingly convection dominated, the predictive accuracy of pure PINNs deteriorates due to stiffness and optimization imbalance. To address this limitation, a hybrid framework is introduced that combines transfer learning with sparse supervision from high fidelity CFD data. The resulting parameterized PINNs model accurately captures the Re dependence of the flow over both interpolation and limited extrapolation regimes while requiring CFD data only over a narrow subset of the parameter space. Detailed comparisons with OpenFOAM simulations demonstrate strong agreement in velocity profiles, and pressure fields. The results show that incorporating governing parameters directly into PINNs enables the construction of parametric Navier Stokes solvers, offering a promising route toward efficient reduced order modeling and data assisted simulation of fluid flows.

💡 Research Summary

The paper presents a novel parameterized Physics‑Informed Neural Network (PINN) framework that treats the Reynolds number (Re) as an explicit input to a single neural network, thereby enabling the model to represent a continuous family of incompressible Navier‑Stokes solutions. The authors demonstrate the approach on the classic two‑dimensional lid‑driven cavity (LDC) problem, which is widely used as a benchmark for viscous flow solvers because its flow topology changes dramatically with Re.

Methodology
A fully‑connected feed‑forward network with ten hidden layers and eighty neurons per layer is employed. The input vector consists of the spatial coordinates (x, y) and a log‑transformed, normalized Re, which compresses the wide dynamic range of Reynolds numbers and aligns well with the tanh activation used in hidden layers. The network outputs the two velocity components (u, v) and pressure (p). Interior collocation points and boundary points are sampled via Monte‑Carlo methods; the physics‑based loss combines a PDE residual term (continuity and momentum equations) with a boundary condition term, weighted by λ_b and λ_PDE. Training proceeds in two stages: first with the Adam optimizer (learning rate decayed from 1e‑3 to 1e‑6) for rapid exploration, then with L‑BFGS for fine‑tuning.

For low Reynolds numbers (Re ≲ 200) the pure PINN—trained solely on the governing equations and boundary conditions—accurately reproduces velocity and pressure fields across the entire training interval and also interpolates well to unseen Re values (e.g., 150, 180). However, as Re increases, convection dominates, the PDE residual becomes stiff, and the pure PINN’s optimization deteriorates.

To overcome this, the authors introduce a hybrid strategy that combines transfer learning with sparse supervision from high‑fidelity CFD data. The network trained at low Re provides initial weights for a second training phase. A small set of CFD snapshots (generated with OpenFOAM on a 128 × 128 grid) at selected high‑Re cases (e.g., 500, 800, 1000) are added as a data‑loss term. This “Hybrid‑PINN” retains the physics‑based loss while being gently guided by a few accurate data points, thereby stabilizing training in convection‑dominated regimes.

Results
The CFD reference solutions are validated against classic benchmark data (Ghia et al., Botella & Peyret, Erturk et al.) and show excellent agreement. For the pure PINN at low Re, the mean absolute error in center‑line velocity profiles is below 0.8 %. In the hybrid regime, the model accurately predicts velocity and pressure for both interpolation (Re = 600, 900) and limited extrapolation (Re = 1100) with errors typically under 1.2 %. Visual comparisons of streamlines, secondary vortices, and pressure contours reveal near‑identical structures to the CFD reference.

Computationally, the parameterized PINN requires roughly 6 × 10⁴ trainable parameters and converges in about eight hours on a single GPU. Once trained, inference for an entire flow field takes ~0.02 s, representing an order‑of‑magnitude speed‑up over conventional CFD (which needs tens of seconds for the same grid and time step).

Discussion and Limitations
The authors acknowledge that the current study is limited to two‑dimensional steady flows. Extending the approach to three‑dimensional, unsteady, or turbulent regimes would require incorporating additional physics (e.g., LES or RANS models) and possibly deeper or multi‑scale network architectures. The log‑normalization of Re works well up to Re ≈ 10³, but for very high Reynolds numbers (>10⁴) the scaling may again become problematic, suggesting the need for hierarchical or adaptive input embeddings. Moreover, the selection of CFD supervision points influences performance; the paper proposes future work on active learning or adaptive sampling to automate this process.

Conclusion
By embedding the governing non‑dimensional parameter directly into the PINN, the authors demonstrate that a single neural network can serve as a parametric Navier‑Stokes solver across a broad Reynolds‑number range. Pure physics‑based training suffices for diffusion‑dominated regimes, while a hybrid transfer‑learning scheme with sparse high‑fidelity data restores accuracy in convection‑dominated regimes. The resulting framework offers high accuracy, substantial computational speed‑up, and the ability to interpolate and modestly extrapolate across unseen flow conditions, positioning it as a promising tool for reduced‑order modeling, digital twins, and real‑time fluid‑flow control applications.