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metadata 
language:
  - en
license: cc-by-4.0
pretty_name: Navier-Stokes Analytical Benchmark
tags:
  - physics
  - fluid-dynamics
  - navier-stokes
  - computational-fluid-dynamics
  - scientific-computing
  - benchmark
  - turbulence
  - compressible-flow
  - non-newtonian
task_categories:
  - other
size_categories:
  - n<1K
annotations_creators:
  - expert-generated
configs:
  - config_name: default
    data_files:
      - split: train
        path: data/train-*
dataset_info:
  features:
    - name: problem_class
      dtype: string
    - name: name
      dtype: string
    - name: description
      dtype: string
    - name: parameters
      dtype: string
    - name: ndim
      dtype: int32
    - name: grid_shape
      sequence:
        dtype: int32
    - name: reynolds_number
      dtype: float64
    - name: time
      dtype: float64
    - name: ux_field
      sequence:
        dtype: float32
    - name: uy_field
      sequence:
        dtype: float32
    - name: uz_field
      sequence:
        dtype: float32
    - name: p_field
      sequence:
        dtype: float32
    - name: rho_field
      sequence:
        dtype: float32
    - name: temperature_field
      sequence:
        dtype: float32
    - name: latex_equation
      dtype: string

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.

Dataset Description

  • Repository: C3S2-Lab/navier-stokes-benchmark
  • Size: 277 samples across 8 problem classes
  • Dimensions: 1D, 2D, and 3D (variable grid_shape)
  • Format: Apache Arrow / Parquet

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.

u(x,t)=e−νtu0(x)\mathbf{u}(\mathbf{x}, t) = e^{-\nu t} \mathbf{u}_0(\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)=U0e−yω/2νcos⁡ ⁣(ωt−yω/2ν)u(y,t) = U_0 e^{-y\sqrt{\omega/2\nu}} \cos\!\left(\omega t - y\sqrt{\omega/2\nu}\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.

dAdt=σA−l∣A∣2A\frac{dA}{dt} = \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}
}