Kolmogorov Flow (Lyapunov-calibrated)

Deterministically forced 2D Navier–Stokes (Kolmogorov flow) at $Re={10}^{4}Re\; =\; 10^4$, sampled at a forecast step calibrated to the system's measured Lyapunov time. Intended for probabilistic / generative neural PDE forecasting on chaotic dynamics.

Generated with JAX-CFD

Summary

• Equation. Vorticity formulation of 2D incompressible Navier–Stokes with deterministic Kolmogorov forcing $f=\sin (k_{f}y)\widehat{x}f\; =\; \backslash sin(k\_f\; y)\backslash ,\backslash hat\; x$ and linear drag.
• Solver. Spectral (jax-cfd, crank_nicolson_rk4), anti-aliased.
• Resolution. $256\times 256256\; \backslash times\; 256$, periodic on $[0,2\pi {]}^2[0,\; 2\backslash pi]^2$.
• Viscosity. $\nu ={10}^{-4}\backslash nu\; =\; 10^\{-4\}$ (Re $\sim {10}^4\backslash sim\; 10^4$.
• Trajectories. 64 independent ICs, 200 saved frames each.
• Format. Single .npy, shape (64, 200, 256, 256), float32.

Time-step calibration

The forecast step $\mathrm{\Delta }\tau \backslash Delta\backslash tau$ between saved frames is set as a fraction of the system's leading Lyapunov time so that consecutive states are dynamically correlated but small initial differences have grown to a macroscopic fraction of the field.

Measured Lyapunov exponent (twin-trajectory, periodic renormalization, after spin-up): $$\mathrm{\Lambda }\approx 1.56{\text{(time}}^{-1}\text{)},T_{\mathrm{\Lambda }}=1\mathrm{/}\mathrm{\Lambda }\approx \mathrm{0.64.}\backslash Lambda\; \backslash approx\; 1.56\backslash ;\backslash ;\backslash text\{(time\}^\{-1\}\backslash text\{)\},\; \backslash qquad\; T\_\backslash Lambda\; =\; 1/\backslash Lambda\; \backslash approx\; 0.64.$$

Solver dt = 1e-3, INNER_STEPS = 100 (solver steps between saved frames), giving: $$\mathrm{\Delta }\tau =100\times {10}^{-3}=0.1,\mathrm{\Delta }\tau \mathrm{/}{T}_{\mathrm{\Lambda }}\approx \mathrm{0.16.}\backslash Delta\backslash tau\; =\; 100\; \backslash times\; 10^\{-3\}\; =\; 0.1,\; \backslash qquad\; \backslash Delta\backslash tau\; /\; T\_\backslash Lambda\; \backslash approx\; 0.16.$$

So consecutive saved frames are separated by roughly one-sixth of a Lyapunov time. Two states that share a coarse-resolution projection but differ in unresolved continuum directions have time to diverge by a factor $e^{0.16}\approx 1.17e^\{0.16\}\; \backslash approx\; 1.17$ between frames.

A warm-up of 5000 solver steps ( $\sim 5{\tau }_{\text{eddy}}\backslash sim\; 5\backslash ,\backslash tau\_\{\backslash text\{eddy\}\}$ is discarded per trajectory before saving, so frames are samples of the statistically stationary regime.

Why this calibration

Probabilistic forecasters that model $p(u_{n+1}\mid u_n)p(u\_\{n+1\}\; \backslash mid\; u\_n)$ as a stochastic interpolant need the conditional to have non-trivial width on the discrete state. If $\mathrm{\Delta }\tau \ll T_{\mathrm{\Lambda }}\backslash Delta\backslash tau\; \backslash ll\; T\_\backslash Lambda$, similar $u_{n}u\_n$ are followed by nearly identical $u_{n+1}u\_\{n+1\}$ and the conditional collapses to a delta — there is no distribution to learn. The choice $\mathrm{\Delta }\tau \sim 0.1\text{–}0.25T_{\mathrm{\Lambda }}\backslash Delta\backslash tau\; \backslash sim\; 0.1\backslash text\{–\}0.25\backslash ,T\_\backslash Lambda$ is the smallest horizon at which chaotic divergence produces a genuinely spread conditional while consecutive frames remain dynamically related (i.e. it is still forecasting, not decorrelated sampling).

Physical setup

Parameter	Value
Domain	$[0,2\pi {]}^2[0,\; 2\backslash pi]^2$, periodic
Grid	$256\times 256256\; \backslash times\; 256$
Viscosity $\nu \backslash nu$	${10}^{-4}10^\{-4\}$
Reynolds	$\sim {10}^4\backslash sim\; 10^4$
Forcing	$f=\sin (4y)\widehat{x}f\; =\; \backslash sin(4y)\backslash ,\backslash hat\; x$ (deterministic)
Drag	$-0.1-0.1$
Solver	jax-cfd spectral, CN-RK4
dt	${10}^{-3}10^\{-3\}$
Inner steps / frame	100
$\mathrm{\Delta }\tau \backslash Delta\backslash tau$	$0.10.1$
$T_{\mathrm{\Lambda }}T\_\backslash Lambda$ (measured)	$\approx 0.64\backslash approx\; 0.64$
Warm-up	5000 solver steps
Trajectories	64
Frames / trajectory	200

Loading

import numpy as np
data = np.load("kolmogorov_Re1e4_64x200x256x256.npy")  # (64, 200, 256, 256), float32

Trajectories are independent (different IC seeds, same physics). Common split: first $N$ trajectories train, remainder validation.

Citation

If you use this dataset, please cite both the underlying solver:

@article{Kochkov2021-ML-CFD,
  author = {Kochkov, Dmitrii and Smith, Jamie A. and Alieva, Ayya and Wang, Qing and Brenner, Michael P. and Hoyer, Stephan},
  title = {Machine learning{\textendash}accelerated computational fluid dynamics},
  volume = {118},
  number = {21},
  elocation-id = {e2101784118},
  year = {2021},
  doi = {10.1073/pnas.2101784118},
  publisher = {National Academy of Sciences},
  issn = {0027-8424},
  URL = {https://www.pnas.org/content/118/21/e2101784118},
  eprint = {https://www.pnas.org/content/118/21/e2101784118.full.pdf},
  journal = {Proceedings of the National Academy of Sciences}
}

@article{Dresdner2022-Spectral-ML,
  doi = {10.48550/ARXIV.2207.00556},
  url = {https://arxiv.org/abs/2207.00556},
  author = {Dresdner, Gideon and Kochkov, Dmitrii and Norgaard, Peter and Zepeda-Núñez, Leonardo and Smith, Jamie A. and Brenner, Michael P. and Hoyer, Stephan},
  title = {Learning to correct spectral methods for simulating turbulent flows},
  publisher = {arXiv},
  year = {2022},
  copyright = {arXiv.org perpetual, non-exclusive license}
}

A method paper accompanying this dataset is in preparation; citation will be added.

License

CC BY 4.0.