C3S2-Lab/navier-stokes-benchmark

Navier-Stokes Analytical Benchmark

A benchmark dataset of fluid dynamics problems with exact or semi-analytical solutions that target structural failure modes of SINDy and EDMD. Designed for evaluating deep-koopman-kan (Koopman-based lifting) and KANDy (equation discovery) pipelines.

Each problem class isolates a specific reason why sparse-regression (SINDy) and linear-Koopman (EDMD) methods provably fail on real Navier-Stokes flows. The latex_equation field serves as the ground-truth reward signal for equation-discovery agents.

Problem Classes

1. ABC Beltrami Flow -- 60 samples

Tri-periodic box $[0, 2\pi]^3$. Beltrami property ($\nabla \times \mathbf{u} = \mathbf{u}$) makes nonlinearity vanish. Exact exponential viscous decay.

$$\mathbf{u}(\mathbf{x},t)=e^{-\nu t}{\mathbf{u}}_0(\mathbf{x})\backslash mathbf\{u\}(\backslash mathbf\{x\},\; t)\; =\; e^\{-\backslash nu\; t\}\; \backslash mathbf\{u\}\_0(\backslash mathbf\{x\})$$

Parameter	Values
$\nu$	0.01, 0.05, 0.1, 0.2
$(A,B,C)$	(1,1,1), (1,0.7,1.3), (0.5,1,1.5)
$t$	0.0, 0.5, 1.0, 2.0, 3.0

2. High-Re Synthetic Turbulence -- 9 samples

Divergence-free random fields with Kolmogorov $E(k) \sim k^{-5/3}$ energy spectrum on a 3D periodic box. SINDy fails: no sparse library exists for cross-scale coupling. EDMD fails: Koopman spectrum is continuous and infinite-dimensional.

Parameter	Values
Re	$10^4$, $5 \times 10^4$, $10^5$
Seeds	3 per Re

3. Oscillating Boundary (Stokes' 2nd Problem) -- 72 samples

Exact solution for flow above an oscillating flat plate. The Stokes layer penetration depth changes with frequency, breaking fixed-domain assumptions. SINDy fails: library defined on a fixed domain. EDMD fails: observable space shifts each cycle.

$$u(y,t)=U_0{e}^{-y\sqrt{\omega \mathrm{/}2\nu }}\cos \text{\hspace{0.17em}⁣}(\omega t-y\sqrt{\omega \mathrm{/}2\nu })u(y,t)\; =\; U\_0\; e^\{-y\backslash sqrt\{\backslash omega/2\backslash nu\}\}\; \backslash cos\backslash !\backslash left(\backslash omega\; t\; -\; y\backslash sqrt\{\backslash omega/2\backslash nu\}\backslash right)$$

Parameter	Values
$U_0$	1.0, 2.0
$\omega$	1.0, 5.0, 10.0
$\nu$	0.01, 0.05, 0.1

4. Hopf Bifurcation (Cylinder Wake) -- 40 samples

Stuart-Landau model of vortex shedding onset near $Re_c \approx 47$. Dynamics change qualitatively at the bifurcation. SINDy fails: coefficients are not constant across the transition. EDMD fails: linear Koopman is provably inadequate at subcritical bifurcations.

$$\frac{dA}{dt}=\sigma A-l\mathrm{\mid }A{\mathrm{\mid }}^{2}A\backslash frac\{dA\}\{dt\}\; =\; \backslash sigma\; A\; -\; l|A|^2\; A$$

Parameter	Values
Re	20, 40, 46, 47, 48, 50, 60, 80, 100, 150
$t$	0, 5, 10, 20

5. Two-Phase Couette Flow -- 18 samples

Exact piecewise-linear velocity with a viscosity discontinuity at the interface. SINDy fails: library cannot represent phase-dependent coefficients. EDMD fails: discontinuities destroy smooth Koopman observables.

Parameter	Values
Interface position $h_1$	0.3, 0.5, 0.7
Viscosity ratio $\mu_2/\mu_1$	0.1, 0.5, 2, 5, 10, 50

6. Turbulent Channel Flow -- 12 samples

Reichardt mean velocity profile with synthetic turbulent fluctuations. SINDy fails: $O(10^6)$ state dimension makes regression underdetermined. EDMD fails: dictionary must grow exponentially with state dimension.

Parameter	Values
$Re_\tau$	180, 395, 590, 1000
Seeds	3 per $Re_\tau$

7. Power-Law (Non-Newtonian) Poiseuille Flow -- 36 samples

Exact analytical solution for shear-thinning and shear-thickening fluids with constitutive law $\tau = K|\dot\gamma|^{n-1}\dot\gamma$. SINDy fails: non-polynomial constitutive relation. EDMD fails: shear-dependent viscosity breaks linear observable assumption.

Parameter	Values
Power-law index $n$	0.3, 0.5, 0.7, 1.0, 1.5, 2.0
Consistency $K$	0.1, 1.0, 5.0
$dP/dx$	-1.0, -5.0

8. Sod Shock Tube (Compressible Euler) -- 30 samples

Exact Riemann solutions for 1D compressible Euler equations with shocks, contact discontinuities, and rarefaction fans. SINDy fails: discontinuities are not polynomial-sparse. EDMD fails: Koopman observables diverge at shock surfaces.

Problem	$(\rho, u, p)_L$	$(\rho, u, p)_R$
Sod	(1, 0, 1)	(0.125, 0, 0.1)
Strong shock	(10, 0, 100)	(1, 0, 1)
Blast	(1, 0, 1000)	(1, 0, 0.01)
Collision	(1, 1, 1)	(1, -1, 1)
Vacuum	(1, -2, 0.4)	(1, 2, 0.4)

Summary: Why SINDy and EDMD Fail

Problem class	SINDy failure mode	EDMD failure mode
High-Re turbulence	Library explodes; no sparse representation	Koopman spectrum is continuous/infinite
Moving boundaries	Fixed basis assumption broken	Observable space non-stationary
Bifurcations	Coefficients not constant	Linear Koopman fails near critical points
Multiphase flows	Phase-dependent coefficients intractable	Discontinuities destroy Koopman linearity
3D wall-bounded turbulence	Curse of dimensionality	Dictionary must grow exponentially
Non-Newtonian fluids	Non-polynomial constitutive law	Shear-dependent viscosity not linear
Compressible shocks	Discontinuities not polynomial-sparse	Koopman observables diverge at shocks

Dataset Schema

Field	Type	Description
problem_class	string	One of 8 problem classes
name	string	Unique sample identifier
description	string	Human-readable description including failure modes
parameters	string (JSON)	All physical parameters
ndim	int32	Spatial dimensionality (1, 2, or 3)
grid_shape	Sequence[int32]	Spatial grid dimensions
reynolds_number	float64	Reynolds number (null if not applicable)
time	float64	Snapshot time
ux_field	Sequence[float32]	x-velocity, flattened
uy_field	Sequence[float32]	y-velocity, flattened (zeros for 1D)
uz_field	Sequence[float32]	z-velocity, flattened (zeros for 1D/2D)
p_field	Sequence[float32]	Pressure field, flattened
rho_field	Sequence[float32]	Density (compressible flows; zeros for incompressible)
temperature_field	Sequence[float32]	Temperature or phase indicator
latex_equation	string	LaTeX governing equations (reward signal)

Usage

from datasets import load_dataset

ds = load_dataset("C3S2-Lab/navier-stokes-benchmark")

# Filter by problem class
shocks = ds["train"].filter(lambda x: x["problem_class"] == "compressible_shock")
turbulence = ds["train"].filter(lambda x: x["problem_class"] == "high_re_turbulence")

# Convert to PyTorch
ds.set_format("torch", columns=["ux_field", "uy_field", "uz_field", "p_field", "rho_field"])

Generate locally

pip install numpy datasets
python generate_ns_dataset.py
python generate_ns_dataset.py --push --repo C3S2-Lab/navier-stokes-benchmark

Intended Use

• Benchmarking agents' fluid mechanics equations discovery.
• Benchmarks are based on the deep-koopman-kan to estimate the lift and KANDy to get the equations.
• Evaluating equation-discovery and symbolic regression methods (via latex_equation)
• Demonstrating structural advantages over SINDy and EDMD on hard N-S problems

Citation

@dataset{c3s2lab_navier_stokes_benchmark,
  title   = {Navier-Stokes Analytical Benchmark},
  author  = {C3S2-Lab},
  year    = {2026},
  url     = {https://huggingface.co/datasets/C3S2-Lab/navier-stokes-benchmark},
  note    = {Fluid dynamics benchmark targeting SINDy/EDMD failure modes}
}