Model Card: PINN for 2D Unsteady Navier–Stokes (Taylor-Green Vortex)

Model Details

• Model type: Physics-Informed Neural Network (PINN); fully-connected MLP surrogate for a PDE solution field
• Architecture: MLP, 6 hidden layers × 64 neurons, Tanh activation, Xavier-normal weight init, zero bias init
  • Input: (x, y, t) — 3 features
  • Output: (u, v, p) — 2D velocity components and pressure
  • Parameters: reported at build time in the notebook (n_params) — record the printed value when re-running
• Framework: PyTorch (run recorded on PyTorch 2.10.0+cu128, CUDA, Tesla T4 GPU)
• Developed by: Modelling and Simulation Laboratory study group, Telkom University
• License / status: Coursework / research exercise notebook — no formal license specified

Intended Use

• Primary use: Educational and research demonstration of PINNs for solving the 2D incompressible unsteady Navier–Stokes equations, using the Taylor-Green Vortex (TGV) as a test case with a known closed-form solution.
• Intended users: Students/researchers exploring PINN methodology; not intended as a production CFD solver.
• Out-of-scope use: Real-world flow prediction, engineering design decisions, or any Reynolds number / geometry outside the trained regime (e.g., turbulent, high-Re, or non-periodic domains). The model is a proof-of-concept validated against a single analytical benchmark and should not be used as a general-purpose Navier–Stokes solver.

Physics / Problem Formulation

• Governing equations: 2D incompressible Navier–Stokes (non-dimensional), momentum-x, momentum-y, and continuity (mass conservation)
• Test case: Taylor-Green Vortex — one of the few 2D Navier–Stokes configurations with an exact analytical solution, used here as ground truth
• Reynolds number: Re = 100 (laminar regime)
• Spatial domain: (x, y) ∈ [0, 2π] × [0, 2π], periodic boundary conditions
• Temporal domain: t ∈ [0, 1]

Training Data

• Collocation points (PDE residual): N_f = 10,000 points sampled uniformly at random over the (x, y, t) domain; re-sampled every 1,000 epochs
• Initial-condition points: N_0 = 2,000 points at t = 0, labeled using the analytical solution
• Boundary points: N_b = 1,000 point-pairs per direction, used to enforce periodic BCs (matching u, v, p at x=0/x=2π and y=0/y=2π)

Training Procedure

• Loss composition: L = λ_pde·L_pde + λ_ic·L_ic + λ_bc·L_bc
  • L_pde: mean-squared PDE residual (momentum-x, momentum-y, continuity)
  • L_ic: mean-squared error vs. analytical u, v at t=0
  • L_bc: mean-squared mismatch between paired periodic boundary points (u, v, p)
  • Weights: λ_pde = 1.0, λ_ic = 10.0 (upweighted as the more fundamental constraint), λ_bc = 1.0
• Optimizer: Adam, initial learning rate 1e-3
• LR schedule: StepLR, step size 5,000 epochs, gamma 0.5
• Epochs: 20,000
• Second-order fine-tuning (e.g., L-BFGS): not used in this run (listed as future work)
• Derivatives: computed via PyTorch autograd (create_graph=True) for PDE residuals up to second order
• Reproducibility: torch.manual_seed(42), np.random.seed(42)
• Compute / wall-clock: ~919.5 seconds (~15.3 minutes) on a Tesla T4 GPU for 20,000 epochs
• Training loss at final logged step (epoch 19,500): total ≈ 4.68e-04 (PDE ≈ 9.78e-05, IC ≈ 2.28e-05, BC ≈ 1.42e-04); loss decreased from ≈9.05 at epoch 0, with some noisy spikes around epochs 3,500–10,000 before settling

Evaluation

• Velocity fields (u, v): relative L2 error stays under ~1.2% across the full time horizon — accurate and stable over time.
• Pressure field (p): raw relative error is large (~93–94%) because pressure in incompressible Navier–Stokes is only defined up to an additive constant (no boundary condition pins its absolute level), and the PINN's output isn't zero-mean by construction. After removing the mean from both predicted and true pressure fields ("pressure gauge" correction), relative error drops to 1.3–2.4%, confirming the shape of the pressure field is learned correctly — only the arbitrary constant offset differs.
• Vorticity: reconstructed via autograd from the velocity output (ω = ∂v/∂x − ∂u/∂y) and compared qualitatively/visually to the analytical vorticity field at t = 0.5; consistent with the analytical field in the notebook's plots.

Limitations

• Pressure ambiguity: raw pressure error is uninformative without gauge correction; any downstream use of p must account for the unknown additive constant.
• Single benchmark, single Re: only validated at Re = 100 on a periodic, doubly-periodic square domain — no evidence of generalization to other Reynolds numbers, geometries, or boundary conditions.
• Training cost: ~15 minutes per run on a T4 GPU for a very simple, well-posed 2D case with a known closed-form solution; substantially slower than classical CFD solvers (FDM/FEM) for equivalent accuracy.
• Convergence stability: training loss showed non-monotonic spikes (e.g., epochs ~3,500, ~4,500, ~10,000) requiring loss monitoring; not guaranteed to converge smoothly with different seeds/hyperparameters.
• Sensitivity: accuracy noted as sensitive to the loss weighting (λ_pde, λ_ic, λ_bc); no ablation over these weights is included.
• No data-driven validation: all "ground truth" comes from the same analytical formula used to construct the loss (IC) and evaluation — this checks self-consistency of the solver, not agreement with real/experimental flow data.

Environmental / Compute Impact

• Single Tesla T4 GPU, ~15.3 minutes training time for the reported run — compute footprint is minor at this scale, but note this is per-run and does not include experimentation/tuning overhead.