README.md

---
license: apache-2.0
task_categories:
  - time-series-forecasting
  - tabular-regression
tags:
  - physics
  - pde
  - fluid-dynamics
  - simulation
  - numerical-methods
size_categories:
  - 10K<n<100K
---

# INC Dataset: Implicit Neural Correction for PDE Solvers

## Dataset Description

This dataset contains simulation data for training and evaluating implicit neural correction methods for partial differential equation (PDE) solvers. The dataset includes two challenging dynamical systems demonstrating complex spatiotemporal behaviors:

1. **Kuramoto-Sivashinsky (KS) Equation** - 1D chaotic dynamics
2. **Backward-Facing Step (BFS) Flow** - 2D incompressible Navier-Stokes with complex geometry

### Dataset Summary

- **Repository**: [INC: Implicit Neural Correction for PDE Solvers](https://github.com/tum-pbs/INC)
- **Paper**: [INC: An Indirect Neural Corrector for Auto-Regressive Hybrid PDE Solvers](https://openreview.net/forum?id=s3Uk3lrfjy) (NeurIPS 2025)

## Dataset Structure

```
INC_Data/
├── KS/
│   └── Dataset/
│       ├── train/
│       ├── valid/
│       └── test/
└── BFS/
    └── Dataset/
    │   ├── train/
    │   ├── valid/
    │   └── test/
    └── Results/
        └── NoModel/        # Baseline without correction
        └── INC_SmallCNN/
            └── {timestamp}_mstep8_.../ # an example model
```
There are two main subdirectories corresponding to the two PDE systems, each containing training, validation, and test datasets. For BFS, there is also a `Results` directory showcasing baseline and corrected model results.

### Data Fields

Each dataset contains time-series simulation data with the following characteristics:


#### Kuramoto-Sivashinsky Equation
- **Spatial Resolution**: 64 grid points
- **Temporal Resolution**: dt = 0.01
- **Parameters**: Periodic boundary conditions
- **File Format**: `.pth` (PyTorch dictionary with trajectories and metadata)
- **Data Structure**: 
  - `trajectories`: shape `(num_trajectories, time_steps, spatial_points)` = `(27, 10001, 64)` for train
  - `domain_size`: shape `(num_trajectories,)` = `(27,)` 
  - `metadata`: dict with generation parameters (gen_dt, resolution, time_scheme, etc.)
- **Dataset Sizes**:
  - Train: 27 trajectories × 10,001 timesteps
  - Valid: 3 trajectories × 10,001 timesteps
  - Test: 6 trajectories × 10,001 timesteps

#### Backward-Facing Step (BFS)
- **Spatial Resolution**: Multi-block grid with refinement, downsampled to approximately $128 \times 32$
- **Temporal Resolution**: Saved with fixed intervals (dt = 0.1)
- **Physical Domain**: 2D channel with backward-facing step geometry (5 blocks)
- **Parameters**: Reynolds numbers {1300, 1350, 1400}, Step height {0.85, 0.875, 1.0}
- **File Format**: Combined `.json` (metadata) + `.npz` (tensor data) per timestep
- **Data Structure**:
  - Each configuration has 5 blocks with varying resolutions
  - Block shapes vary by position: e.g., `(1, 2, 16, 16)` for velocity, `(1, 1, 16, 16)` for pressure
  - Metadata includes: domain name, spatial dimensions, viscosity, block info, boundaries
  - Data arrays: velocity (2 channels), pressure (1 channel), vertex coordinates, boundary conditions
- **Dataset Sizes**:
  - Train: 3 configurations × ~801 timesteps each
  - Valid: 1 configuration × 301 timesteps
  - Test: 1 configuration × 3,001 timesteps


## Dataset Generation

The data was generated using classical numerical PDE solvers:

- **Burgers**: 5th-order WENO scheme with RK4 time integration
- **Kuramoto-Sivashinsky**: Pseudo-spectral method with exponential time differencing
- **BFS**: PISO algorithm with custom CUDA implementation for multi-block domains

### Generation Scripts

The original data generation code is available in the [INC repository](https://github.com/tum-pbs/INC):
- `scripts/Sim_BFS.py` - Generate BFS flow data
- `solvers/solver_1d.py` - Contains Burgers and KS solvers

## Use Cases

This dataset is designed for:

1. **Hybrid Physics-ML Models**: Training neural networks to correct numerical solver errors
2. **Operator Learning**: Learning mappings between PDE solution spaces
3. **Time-Series Forecasting**: Predicting long-term evolution of chaotic dynamical systems
4. **Benchmarking**: Evaluating neural PDE solver architectures (FNO, U-Net, DeepONet)
5. **Physics-Informed Learning**: Combining data-driven and physics-based approaches


See the [paper](https://openreview.net/forum?id=s3Uk3lrfjy) for detailed results and methodology.

## Citation

If you use this dataset in your research, please cite:

```bibtex
@article{INC2025,
  title={{INC}: An Indirect Neural Corrector for Auto-Regressive Hybrid {PDE} Solvers},
  author={Hao Wei, Aleksandra Franz, Björn Malte List, Nils Thuerey},
  booktitle={The Thirty-ninth Annual Conference on Neural Information Processing Systems},
  year={2025},
  url={https://openreview.net/forum?id=s3Uk3lrfjy}
}
```

## Limitations and Biases

- **Domain Specificity**: Dataset is limited to three specific PDEs; generalization to other equations may require additional data
- **Resolution Trade-off**: Coarser resolutions improve computational efficiency but may miss fine-scale features
- **Boundary Conditions**: Limited to periodic (KS) and no-slip wall (BFS) boundaries
- **Parameter Range**: Limited range of physical parameters (viscosity, Reynolds number, domain geometry)

## Additional Information

### Licensing

This dataset is released under the Apache 2.0 License. You are free to use, modify, and distribute the data with proper attribution.

### Contact

For questions or issues with the dataset:
- **GitHub Issues**: [INC Repository](https://github.com/tum-pbs/INC/issues)

### Acknowledgments

This work builds upon numerical methods and deep learning architectures from:
- PICT solver (Franz et al., 2025)
- Fourier Neural Operator (Li et al., 2020)
- DeepONet (Lu et al., 2021)


---

**For detailed usage instructions and training examples, see the [main repository](https://github.com/tum-pbs/INC).**