Taylor–Green Vortex

The classic 3D transition-to-turbulence benchmark at $\mathrm{Re}=1600$: incompressible Navier–Stokes on the periodic cube $[0,2\pi {]}^3$,

$$\frac{\partial \mathbf{u}}{\partial t}+(\mathbf{u}\cdot \mathrm{\nabla })\mathbf{u}+\mathrm{\nabla }p-\nu {\mathrm{\nabla }}^2\mathbf{u}=0,\mathrm{\nabla }\cdot \mathbf{u}=0,$$

starting from the analytic vortex ${\mathbf{u}}_0=(\cos x\sin y\cos z,-\sin x\cos y\cos z,0)$. The initial vortex sheet stretches, rolls up, and cascades to small scales.

Initial z-vorticity on the z = 0 plane.

Run

cd examples/taylor_green

# Standard time-window training
python3 main.py --config=configs/baseline.py

# Multi-stage eps-homotopy training
python3 main.py --config=configs/multi_stage.py

Config	Description
configs/baseline.py	Single network per time window
configs/multi_stage.py	Multi-stage correction cascade per window
configs/pseudo_time.py	Adds adaptive pseudo-time stepping

Notes

• The flagship example for multi-stage training: each stage trains a correction network around the frozen previous stages ($\text{sol}=\text{prev}+\epsilon \text{diff}$) with a linearized Navier–Stokes residual, a larger Fourier frequency, and rejection-sampled collocation points where the previous residual is large.
• Long cascades are fully resumable: per-window, per-stage checkpoints (time_window_{w}_stage_{s}) plus --config.training.resume=True.
• Enstrophy of the predicted field is logged at every window boundary as a physical sanity check against the known Re 1600 reference curve.