M³: Multi-scale Measure Modeling

Official reference implementation of the three-stage M³ pipeline.

Paper: M³: Reframing Training Measures for Discretized Physical Simulations
arXiv: 2605.08843 · pdf

M³ constructs a training measure for discretized physical simulations via:

1. Partition — variation-aware Morton octree
2. Grouping — multi-scale strata
3. Sampling — inter-level budget + water-filling + within-cell sampling

Install

pip install numpy
export PYTHONPATH=$PWD   # from repo root

Quick start

python examples/toy_example.py

from m3 import run_m3

out = run_m3(
    points_xyz=xyz,      # (N, 3)
    phi_scalar=pressure, # (N,)
    psi_vec3=wss,        # (N, 3) or None
    budget=8192,
    file_type="boundary",  # or "volume"
    seed=42,
    rho_fill_ratio=0.3,
)
# out.sampled_indices  → indices into the original point array

Hyperparameters: hyperparameters.yaml.

Layout

M3/
├── hyperparameters.yaml
├── m3/
│   ├── pipeline.py
│   ├── step1_partitioning.py
│   ├── step2_grouping.py
│   └── step3_sampling.py
└── examples/toy_example.py

Citation

@article{mei2026m3,
  title={M$^3$: Reframing Training Measures for Discretized Physical Simulations},
  author={Mei, Yuan and Song, Xiaowen and Song, Xingjian and Takeishi, Naoya},
  journal={arXiv preprint arXiv:2605.08843},
  year={2026}
}

License

MIT (see LICENSE).