README.md

---
license: mit
task_categories:
- other
pretty_name: PreGen Navier-Stokes 2D Dataset
size_categories:
- 100K<n<1M
tags:
- physics
- fluid-dynamics
- navier-stokes
- pde
- scientific-computing
- neural-operators
- foundation-models
- difficulty-transfer
- reynolds-number
- openfoam
---

# PreGen Navier-Stokes 2D Dataset

## Dataset Description

This dataset accompanies the research paper **"Pre-Generating Multi-Difficulty PDE Data For Few-Shot Neural PDE Solvers"** (under review at ICLR 2026). It contains systematically generated 2D incompressible Navier-Stokes fluid flow simulations designed to study **difficulty transfer** in neural PDE solvers.

The key insight: by pre-generating many low and medium difficulty examples and including them with a small number of hard examples, neural PDE solvers can learn high-difficulty physics from far fewer samples. This dataset enables **8.9× reduction in compute time** while achieving comparable performance.

### Dataset Summary

- **Total Size:** ~421 GB
- **Format:** NumPy arrays (.npy files)
- **Number of Files:** 9
- **Simulations per file:** 6,400 trajectories
- **Timesteps:** 20 per trajectory
- **Spatial Resolution:** 128 × 128 grid
- **Solver:** OpenFOAM (icoFoam)
- **Domain:** 2D Incompressible Navier-Stokes equations

### Problem Setting

The dataset solves the 2D incompressible Navier-Stokes equations:

```
∂u/∂t + (u · ∇)u + ∇p = ν∆u
∇ · u = 0
```

where:
- `u(x,t)` is the velocity field
- `p(x,t)` is the kinematic pressure
- `ν` is the kinematic viscosity (1.5 × 10⁻⁵ m²/s)
- Domain: Ω ⊂ [0,1]²

## Difficulty Axes

The dataset systematically varies complexity along three axes:

### 1. **Geometry Axis** (Number of Obstacles)
Simulations in flow-past-object (FPO) configuration with varying obstacle complexity:

- **Easy:** No obstacles (open channel flow)
- **Medium:** Single square obstacle
- **Hard:** 2-10 randomly placed square obstacles

**Files:**
- `Geometry_Axis/FPO_Geometry_Easy_NoObstacle.npy` (47 GB)
- `Geometry_Axis/FPO_Geometry_Medium_SingleObstacle.npy` (47 GB)
- `Geometry_Axis/FPO_Geometry_Hard_MultiObstacle.npy` (47 GB)

### 2. **Physics Axis** (Reynolds Number)
Simulations with varying flow complexity via Reynolds number:

**Multi-Obstacle Flows:**
- **Easy:** Re ∈ [100, 1000] - laminar regime
- **Medium:** Re ∈ [2000, 4000] - transitional regime
- **Hard:** Re ∈ [8000, 10000] - turbulent regime

**Files:**
- `Physics_Axis/MultiObstacle/FPO_Physics_MultiObstacle_Easy_Re100-1000.npy` (47 GB)
- `Physics_Axis/MultiObstacle/FPO_Physics_MultiObstacle_Medium_Re2000-4000.npy` (47 GB)
- `Physics_Axis/MultiObstacle/FPO_Physics_MultiObstacle_Hard_Re8000-10000.npy` (47 GB)

**No-Obstacle Flows:**
- `Physics_Axis/NoObstacle/FPO_Physics_NoObstacle_Easy_Re100-1000.npy` (47 GB)

### 3. **Combined Axis** (Geometry + Physics)
Combined variations in both geometry and Reynolds number:

- **Easy:** No obstacles + low Re ([100, 1000])
- **Medium:** Single obstacle + medium Re ([2000, 4000])
- **Hard:** Multiple obstacles + high Re ([8000, 10000])

**File:**
- `Combined_Axis/FPO_Combined_Medium_SingleObstacle_MedRe.npy` (47 GB)

### 4. **Special Configuration**
- `Special/FPO_Cylinder_Hole_Location_6284.npy` (47 GB) - Cylinder with hole at specific location

## Data Format

Each `.npy` file contains a NumPy array with shape: `(6400, 20, 128, 128, 6)`

**Dimensions:**
- **6400**: Number of simulation trajectories
- **20**: Timesteps per trajectory
- **128 × 128**: Spatial grid resolution
- **6**: Channels (features)

**Channels (in order):**
1. **u** - Horizontal velocity component (m/s)
2. **v** - Vertical velocity component (m/s)
3. **p** - Kinematic pressure (m²/s²)
4. **Re_normalized** - Normalized Reynolds number
5. **Binary mask** - Geometry encoding (1 = obstacle, 0 = fluid)
6. **SDF** - Signed distance field to nearest obstacle boundary

## Simulation Details

### Boundary Conditions

**Flow Past Object (FPO):**
- **Left (inlet):** Parabolic velocity profile with peak velocity Umax
- **Right (outlet):** Zero-gradient pressure outlet
- **Top/Bottom:** No-slip walls (u = 0)
- **Obstacles:** No-slip walls (u = 0)

### Reynolds Number Sampling
Re is sampled from a truncated Gaussian distribution N(5000, 2000²) with support [100, 10000]. The inlet velocity is scaled to achieve the target Re:

```
Re = (U_avg × L) / ν
U_avg = (2/3) × U_max
```

### Time Integration
- **Scheme:** Backward Euler (1st order implicit)
- **Spatial discretization:** Finite volume method
- **Gradient terms:** Gauss linear (central differencing)
- **Convection:** Gauss linearUpwind with gradient reconstruction
- **Diffusion:** Gauss linear orthogonal

### Simulation Duration
Adaptive time scheduling based on Reynolds number to ensure flow development:
- **Low Re (10-100):** Fixed 2700s
- **Medium Re (100-1000):** 1-10× characteristic diffusion time
- **High Re (1000-10000):** 10-40× characteristic diffusion time

### Computational Cost
The harder the simulation, the more expensive to generate:

| Configuration | Average Time (seconds) |
|--------------|----------------------|
| No obstacle, Low Re | 176.7 |
| No obstacle, Medium Re | 261.1 |
| No obstacle, High Re | 350.4 |
| One obstacle, Low Re | 609.5 |
| One obstacle, Medium Re | 731.1 |
| One obstacle, High Re | 942.8 |
| Multiple obstacles, Low Re | 1550.9 |
| Multiple obstacles, Medium Re | 1599.2 |
| Multiple obstacles, High Re | 1653.3 |

## Key Research Findings

This dataset was specifically designed to study **difficulty transfer** in neural PDE solvers:

1. **Sample Efficiency**: Training on 10% hard data + 90% easy/medium data recovers ~96-98% of the performance of training on 100% hard data

2. **Compute Efficiency**: By mixing difficulties optimally, you can achieve the same error with **8.9× less compute** spent on data generation

3. **Medium > Easy**: For most budgets, generating fewer medium-difficulty examples outperforms generating more easy examples

4. **Foundation Dataset Potential**: Medium-difficulty data (single obstacle) improves few-shot performance on complex geometries (NURBS shapes from FlowBench)

## Usage

### Basic Loading

```python
import numpy as np
from huggingface_hub import hf_hub_download

# Download a specific difficulty level
file_path = hf_hub_download(
    repo_id="sage-lab/PreGen-NavierStokes-2D",
    filename="Geometry_Axis/FPO_Geometry_Easy_NoObstacle.npy",
    repo_type="dataset"
)

# Load the data
data = np.load(file_path)
print(f"Data shape: {data.shape}")  # (6400, 20, 128, 128, 6)

# Extract individual trajectories
trajectory_0 = data[0]  # Shape: (20, 128, 128, 6)

# Extract velocity and pressure
u = trajectory_0[:, :, :, 0]  # Horizontal velocity
v = trajectory_0[:, :, :, 1]  # Vertical velocity
p = trajectory_0[:, :, :, 2]  # Pressure
mask = trajectory_0[:, :, :, 4]  # Binary geometry mask
sdf = trajectory_0[:, :, :, 5]  # Signed distance field
```

### Difficulty Mixing for Training

```python
import numpy as np
from huggingface_hub import hf_hub_download

# Load different difficulty levels
easy_data = np.load(hf_hub_download(
    repo_id="sage-lab/PreGen-NavierStokes-2D",
    filename="Geometry_Axis/FPO_Geometry_Easy_NoObstacle.npy",
    repo_type="dataset"
))

medium_data = np.load(hf_hub_download(
    repo_id="sage-lab/PreGen-NavierStokes-2D",
    filename="Geometry_Axis/FPO_Geometry_Medium_SingleObstacle.npy",
    repo_type="dataset"
))

hard_data = np.load(hf_hub_download(
    repo_id="sage-lab/PreGen-NavierStokes-2D",
    filename="Geometry_Axis/FPO_Geometry_Hard_MultiObstacle.npy",
    repo_type="dataset"
))

# Recommended: Use 10% hard + 90% medium for cost-effective training
n_hard = 80
n_medium = 720

train_data = np.concatenate([
    hard_data[:n_hard],
    medium_data[:n_medium]
], axis=0)

# Hold out 100 hard examples for testing
test_data = hard_data[-100:]
```

### Computing Metrics

```python
def compute_nmae(y_true, y_pred):
    """
    Compute normalized Mean Absolute Error (nMAE)
    as used in the paper.

    Args:
        y_true: Ground truth, shape (N, T, H, W, C)
        y_pred: Predictions, shape (N, T, H, W, C)

    Returns:
        nMAE: Normalized mean absolute error
    """
    numerator = np.abs(y_true - y_pred).sum()
    denominator = np.abs(y_true).sum()
    return numerator / (denominator + 1e-10)
```

## Tested Models

The paper evaluates this dataset on:

### Supervised Neural Operators (trained from scratch)
- **CNO** (Convolutional Neural Operator) - 18M parameters
- **F-FNO** (Factorized Fourier Neural Operator) - 5-layer

### Foundation Models (fine-tuned)
- **Poseidon-T** (Tiny) - 21M parameters
- **Poseidon-B** (Base) - 158M parameters
- **Poseidon-L** (Large) - 629M parameters

All models are trained autoregressively with one-step-ahead prediction (t → t+1) using relative L1 loss.

## Citation

If you use this dataset, please cite:

```bibtex
@inproceedings{pregen2026,
  title={Pre-Generating Multi-Difficulty {PDE} Data For Few-Shot Neural {PDE} Solvers},
  author={Anonymous},
  booktitle={Under review at International Conference on Learning Representations (ICLR)},
  year={2026},
  url={https://openreview.net}
}
```

**Note:** Citation will be updated once the paper is published.

## Related Datasets

- **The Well** - Large-scale multi-physics PDE dataset
- **PDEBench** - Benchmark for scientific machine learning
- **FlowBench** - Flow simulation over complex geometries (NURBS shapes)

## License

MIT License

## Acknowledgments

This dataset was generated using:
- **OpenFOAM** (v2406) for CFD simulations
- Simulations performed on computational clusters
- Total compute time: Several thousand GPU/CPU hours

## Contact

For questions or issues:
- Open an issue in the dataset repository
- Contact the sage-lab organization on Hugging Face
- See the paper for additional contact information (once published)

## Dataset Maintainers

sage-lab organization

---

**Dataset Version:** 1.0
**Last Updated:** 2024
**Status:** Research dataset under peer review