PINN Navier-Stokes Solver (Kovasznay + Cylinder Flow)

This repository contains Physics-Informed Neural Networks (PINNs) for two incompressible Navier-Stokes benchmarks:

• Steady 2D Kovasznay flow
• Unsteady 2D cylinder wake flow

The models learn velocity and pressure fields by minimizing boundary supervision loss and PDE residual loss.

Training Results

Kovasznay Flow (Re = 20)

Best run metrics captured from logged training summaries:

• Boundary MSE: 1.20e-09
• Physics MSE: 2.24e-07
• Final weighted objective: -15.63
• RAR + L-BFGS run: Boundary MSE 1.14e-09, Physics MSE 2.81e-07

Interpretation:

• The PINN converges to very low boundary and physics residual errors.
• Adaptive weighting and second-stage optimization (L-BFGS) are critical to reach the final regime.
• Residual-based adaptive refinement (RAR) stabilizes physics loss as collocation density increases in hard regions.

Cylinder Flow (Re = 100)

Representative logged run metrics:

• Boundary MSE: 3.67e-06
• Physics MSE: 1.45e-05
• Final weighted objective: -9.53

Interpretation:

• The unsteady wake problem is more challenging than Kovasznay; residuals remain higher, which is expected.
• The pipeline supports additional RAR passes to improve wake-region fidelity.

Result Figures

Kovasznay Field Predictions

Cylinder Flow Progression

Governing Equations

For velocity $(u, v)$ and pressure $p$, the incompressible Navier-Stokes equations are:

$$ \nabla \cdot \mathbf{u} = 0 $$

$$ \mathbf{u} \cdot \nabla \mathbf{u} + \nabla p - \frac{1}{Re} \nabla^2 \mathbf{u} = 0 $$

Cylinder flow extends this with the transient term $\partial \mathbf{u}/\partial t$.

Project Structure

PDE-Solver/
├── README.md
├── requirements.txt
├── pinn_kovasznay.pth
├── pinn_cylinder.pth
├── kovasznay_flow/
│   ├── dataset.py
│   ├── evaluate.py
│   ├── kovasznay.py
│   ├── loss.py
│   ├── network.py
│   ├── refine.py
│   ├── train.py
│   └── results/
└── cylinder_flow/
    ├── dataset.py
    ├── evaluate.py
    ├── loss.py
    ├── network.py
    ├── refine.py
    ├── train.py
    └── results/

Installation

pip install -r requirements.txt

Usage

Run all commands from the repository root.

Kovasznay

python kovasznay_flow/train.py
python kovasznay_flow/refine.py
python kovasznay_flow/evaluate.py

Cylinder

python cylinder_flow/train.py
python cylinder_flow/refine.py
python cylinder_flow/evaluate.py

Notes

• Scripts automatically use GPU if CUDA is available; otherwise they run on CPU.
• Existing plots in */results/ reflect prior training runs and can be overwritten by new experiments.
• Weights & Biases logging is enabled in training/refinement scripts.

License

This project is distributed under the terms in LICENSE.