---
language:
- en
license: mit
arxiv: 2506.08604
task_categories:
- other
tags:
- physics
- pde
- flow-matching
- scientific-machine-learning
---

# Physics vs Distributions: Pareto Optimal Flow Matching with Physics Constraints

<div>
  
  [![arXiv](https://img.shields.io/badge/arXiv-2506.08604-b31b1b.svg)](https://arxiv.org/abs/2506.08604)
  [![View on GitHub](https://img.shields.io/badge/GitHub-Official%20Code-181717?logo=github)](https://github.com/tum-pbs/PBFM)

</div>

This repository contains the datasets for the **dynamic stall** and **Kolmogorov flow** cases presented in the paper "[Physics vs Distributions: Pareto Optimal Flow Matching with Physics Constraints](https://huggingface.co/papers/2506.08604)".

## Dynamic Stall dataset
The design space is defined as a four-dimensional hypercube. The design variables are:
| Design variable         | Symbol                                 | Description                                          | Range       |
|-------------------------|----------------------------------------|------------------------------------------------------|-------------|
| Free-stream Mach number | $ M_{\infty} $                         | Ratio of free-stream velocity to speed of sound      | 0.3 – 0.5   |
| Mean angle of attack    | $ \alpha_0 $                           | Average angle between chord line and flow direction  | 5° – 10°    |
| Pitching amplitude      | $ \alpha_s $                           | Maximum angular deviation during pitching motion     | 5° – 10°    |
| Reduced frequency       | $ k = \dfrac{\omega c}{2V_{\infty}} $  | Non-dimensional frequency of oscillation             | 0.05 – 0.1  |

The hypercube is sampled with **128 points for training** and **16 points for testing**. Each sampled point represents a nominal operating condition.

Each nominal condition is perturbed as follows:

$$
x_{\text{perturbed}} = (1 + \mathcal{N}(0, 0.02)) \cdot x_{\text{nominal}}
$$

where $\mathcal{N}(0, 0.02)$ denotes a Gaussian noise term with zero mean and standard deviation 0.02.

This results in **32 perturbed variations per nominal condition**, yielding a total of:

- $128 \times 32 = 4096$ simulations for training
- $16 \times 32 = 512$ simulations for testing

Each simulation that corresponds to a dataset sample has 6 fields of size $128 \times 128$. The fields correspond to:
- Absolute pressure
- x-wall tangential velocity gradient
- y-wall tangential velocity gradient
- Temperature
- Density
- Wall shear stress

Each `hdf5` file contains three arrays:
- `fields` with shape `(conditions, samples per condition, fields, x, y)`
- `nominal_condition` with shape `(nominal conditions, samples per condition, design variables)`
- `real_condition` with shape `(real conditions, samples per condition, design variables)`

## Kolmogorov flow dataset
The Kolmogorov flow problem spans Reynolds numbers
in the range $[100, 500]$, using a spatial resolution of $128 \times 128$. The simulations are performed using [TorchFSM](https://zenodo.org/records/15350210). The **training dataset includes 32 different flow conditions**, while the **validation dataset contains 16 conditions**. Each condition has $1\, 024$ snapshots.

Each simulation that corresponds to a dataset sample has 2 fields of size $128 \times 128$. The fields correspond to:
- x-velocity
- y-velocity

Each `hdf5` file contains two arrays:
- `fields` with shape `(conditions, samples per condition, fields, x, y)`
- `reynolds` with shape `(reynolds numbers, )`

## Citation

```bibtex
@inproceedings{pbfm2026,
    title={Physics vs Distributions: Pareto Optimal Flow Matching with Physics Constraints},
    author={Giacomo Baldan and Qiang Liu and Alberto Guardone and Nils Thuerey},
    booktitle={The Fourteenth International Conference on Learning Representations},
    year={2026},
    url={https://openreview.net/forum?id=tAf1KI3d4X}
}
```