Kolmogorov Flow

2D incompressible Navier–Stokes on a periodic square, driven by a sinusoidal body force — the flow is unsteady and vortical:

$$\frac{\partial \mathbf{u}}{\partial t}+(\mathbf{u}\cdot \mathrm{\nabla })\mathbf{u}+\mathrm{\nabla }p-\nu {\mathrm{\nabla }}^2\mathbf{u}=\mathbf{f},\mathrm{\nabla }\cdot \mathbf{u}=0,\mathbf{f}=(2\sin (4\pi y),0).$$

Results

vorticity error 4.2e-02recipe 4 time windows + fixed pseudo-time4 × 100k steps, single GPU

The vortical dynamics are tracked over the full horizon to a relative vorticity error of 4.2e-02 (stitched across all four windows), versus 0.59 when trained as a single global fit — the time-window curriculum is the decisive ingredient for this flow. Pseudo-time stepping composes cleanly with it: constant damping weights (step size 1.0) train stably through every window, and an adaptive-weight variant matches it (4.4e-02 even without windows in the single-window configuration). The animation shows the prediction staying phase-locked with the reference as vortices merge and stretch; the error concentrates in the thin filaments between vortices.

Reference, PINN prediction, and absolute error of the vorticity field over all four time windows.

Training losses and per-variable errors across the four windows (each window warm-starts from the previous one's parameters).

Run

cd examples/kolmogorov_flow
python3 main.py --config=configs/pseudo_time.py \
    --config.pseudo_time.strategy=constant \
    --config.training.num_time_windows=4

Config	Description
configs/baseline.py	PirateNet + causal weighting
configs/pseudo_time.py	Adaptive pseudo-time stepping
configs/fixed_pseudo_time.py	Constant pseudo-time weights

Notes

• Uses time-window training with transfer learning between windows; the vorticity is computed from the velocity network by automatic differentiation ($\omega =v_x-u_y$).
• The higher-Reynolds version of this problem is the Kolmogorov flow at Re 10⁶.