thuerey-group/PBFM

Physics vs Distributions: Pareto Optimal Flow Matching with Physics Constraints

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".

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:

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

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. 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

@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}
}