---
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](https://huggingface.co/datasets/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.

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

| 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.

$$u(y,t) = U_0 e^{-y\sqrt{\omega/2\nu}} \cos\!\left(\omega t - y\sqrt{\omega/2\nu}\right)$$

| 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.

$$\frac{dA}{dt} = \sigma A - l|A|^2 A$$

| 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

```python
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

```bash
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

```bibtex
@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}
}
```