Kolmogorov Flow at Re 10⁶

The showcase problem: forced 2D turbulence at a viscosity a thousand times smaller than the standard benchmark, trained forward in time from a DNS snapshot:

$$\frac{\partial \mathbf{u}}{\partial t}+(\mathbf{u}\cdot \mathrm{\nabla })\mathbf{u}+\mathrm{\nabla }p-\nu \mathrm{\Delta }\mathbf{u}=(0.1\sin (4\pi y),0),\mathrm{\nabla }\cdot \mathbf{u}=0,\nu ={10}^{-6}.$$

DNS reference vorticity on a 2048² grid.

Run

The DNS reference data (~1.3 GB) is not tracked in git; place it under data/ or point the config at it (see data/README.md):

cd examples/kolmogorov_flow_Re1e6
python3 main.py --config=configs/baseline.py \
    --config.data_path=/path/to/kolmogorov_flow_Re1e6.npy

Config	Description
configs/baseline.py	PirateNet + causal weighting, time windows
configs/pseudo_time.py	Adds adaptive pseudo-time stepping

Notes

• Time-window training with resume: --config.training.resume=True continues from the last trained window after an interruption.
• Initial conditions for each window are predicted on the full 2048² grid in batches (predict_in_batches), using the schedule-free averaged parameters.
• Enstrophy at each window boundary is logged as a physical sanity check.