Cahn–Hilliard Phase Separation (cahn-hilliard)

One line description of the data: 576 Cahn–Hilliard phase-separation simulations on the sphere, with 64 snapshots on a $256\times 512256\; \backslash times\; 512$ latitude–longitude grid.

Longer description of the data: The Cahn–Hilliard equation describes phase separation in binary mixtures. Starting from a nearly uniform field with small Gaussian perturbations, the system undergoes spinodal decomposition into two phases and subsequently coarsens into progressively larger domains. We vary the interface width $\epsilon \backslash varepsilon$, the mean initial composition, the initial noise variance, and the sphere radius. The dataset provides a spherical benchmark with fourth-order spatial dynamics and a globally conserved order parameter.

Code or software used to generate the data: FiPy, using the NIST examples/cahnHilliard/sphere.py setup as the reference implementation.

Visualization:

About the data

Dimension of discretized data: 64 snapshots per trajectory, at $t\approx 0,10,\dots ,620t\; \backslash approx\; 0,\; 10,\; \backslash ldots,\; 620$ and $630630$, on a $256\times 512256\; \backslash times\; 512$ (lat × lon) grid.

Fields available in the data:

HDF5 path	Shape	Units	Description
t0_fields/phi	(1, 64, 256, 512)	dimensionless	Order parameter $\phi \backslash varphi$; the two coexisting phases correspond approximately to $\phi =0\backslash varphi\; =\; 0$ and $\phi =1\backslash varphi\; =\; 1$
scalars/{epsilon, mean_init, variance, radius, seed, D, a}	()	—	Run parameters (stored as floats)
dimensions/{time, lat, lon}	(64,), (256,), (512,)	—, °, °	Coordinates

The stored field is float32; the time coordinate is float64. The $t=0t\; =\; 0$ snapshot is the raw noise initial condition and can extend well outside $[0,1][0,1]$ (in runs with variance = 0.05, from about $-0.35-0.35$ to $1.441.44$). After that, $\phi \backslash varphi$ stays within about 0.02 of $[0,1][0,1]$.

Number of trajectories: 576: $4\times 3\times 4\times 34\; \backslash times\; 3\; \backslash times\; 4\; \backslash times\; 3$ parameter combinations, each with 4 seeds. 461/58/57 train/validation/test split in data/{train,valid,test}/. Run IDs are ordered by parameter with epsilon outermost, so the first 17 test runs all have $\epsilon =0.5\backslash varepsilon\; =\; 0.5$: sample across a split rather than taking its first $NN$ runs.

Estimated size of the ensemble of all simulations: Approximately 15 GB (gzip-compressed HDF5).

Storage format: One HDF5 file per run. Normalization statistics computed on the train split are in stats.yaml.

Grid type: $256\times 512256\; \backslash times\; 512$ regular equiangular latitude–longitude grid, excluding the poles. The simulations are performed on an unstructured spherical Gmsh mesh and resampled to the regular grid using inverse-distance weighting over the four nearest mesh cells.

Initial conditions: Uncorrelated Gaussian noise, drawn independently for each cell of the native mesh, with mean mean_init and variance variance. No spin-up is discarded.

Boundary conditions: Spherical.

Simulation time-step $(\mathrm{\Delta }t)(\backslash Delta\; t)$: Adaptive. The time step starts at $\exp (-5)\approx 0.0067\backslash exp(-5)\; \backslash approx\; 0.0067$ solver-time units, and grows by a factor $e^{0.01}e^\{0.01\}$ per step. The configured cap of 100 is never reached within $t\le 630t\; \backslash le\; 630$; the step only reaches about 6.2.

Data are stored separated by $(\delta t)(\backslash delta\; t)$: ≈10 solver-time units. Snapshot $ii$ is written at the first solver step with $t\ge 10it\; \backslash ge\; 10i$, so it lags its mark by up to 5.8 units, and the spacing varies between 4.2 and 13.4. The final snapshot is at exactly $t=630t\; =\; 630$. The time-step schedule does not depend on the run parameters, so all runs share the same dimensions/time.

Total time range $(t_{\text{min}}(t\_\backslash textup\{min\}$ to $t_{\text{max}})t\_\backslash textup\{max\})$: 0 to 630 solver-time units.

Set of coefficients or non-dimensional parameters evaluated:

• epsilon $=\epsilon \in \{0.5,1.0,1.5,2.0\}=\; \backslash varepsilon\; \backslash in\; \backslash \{0.5,\; 1.0,\; 1.5,\; 2.0\backslash \}$, the interface-width parameter
• mean_init $\in \{0.35,0.50,0.65\}\backslash in\; \backslash \{0.35,\; 0.50,\; 0.65\backslash \}$, the mean initial composition
• variance $\in \{0.001,0.005,0.01,0.05\}\backslash in\; \backslash \{0.001,\; 0.005,\; 0.01,\; 0.05\backslash \}$, the initial noise variance
• radius $=R\in \{5.0,7.5,10.0\}=\; R\; \backslash in\; \backslash \{5.0,\; 7.5,\; 10.0\backslash \}$, the sphere radius
• seed $\in \{0,1,2,3\}\backslash in\; \backslash \{0,\; 1,\; 2,\; 3\backslash \}$, the initial-condition seed

The mobility and bulk free-energy scale are fixed to $D=1D\; =\; 1$ and $a=1a\; =\; 1$.

Approximate time to generate the data: Approximately 5–30 minutes per trajectory, depending mainly on the sphere radius.

Hardware used to generate the data: CPU generation on sciCORE compute nodes (AMD EPYC or Intel Xeon), using one process per run.

Loading

Download a single run (≈29 MB) and read it with h5py:

import h5py
from huggingface_hub import hf_hub_download

path = hf_hub_download("sada-group/cahn-hilliard", "data/test/run_0001.h5", repo_type="dataset")
with h5py.File(path, "r") as f:
    phi = f["t0_fields/phi"][0]                              # (64, 256, 512)
    t = f["dimensions/time"][:]                              # solver time; spacing is not exactly 10
    params = {k: f["scalars"][k][()] for k in f["scalars"]}

To download a whole split (or the full 15 GB dataset), use snapshot_download with allow_patterns:

from huggingface_hub import snapshot_download

root = snapshot_download(
    "sada-group/cahn-hilliard",
    repo_type="dataset",
    allow_patterns=["data/test/*", "stats.yaml"],  # drop allow_patterns for everything
)

Please cite the associated paper if you use this data in your research:

@misc{muser2026dandelionsphericalflowerneural,
      title={Dandelion: A Spherical Flower for Neural Simulation of Planetary Dynamics},
      author={Till Muser and Giovanni Abati and Ivan Dokmanić},
      year={2026},
      eprint={2608.27521},
      archivePrefix={arXiv},
      primaryClass={cs.LG},
      url={https://arxiv.org/abs/2608.27521},
}

as well as the original Cahn–Hilliard references and the solver:

@article{cahn1958free,
  title={Free Energy of a Nonuniform System. I. Interfacial Free Energy},
  author={Cahn, John W. and Hilliard, John E.},
  journal={The Journal of Chemical Physics},
  volume={28},
  number={2},
  pages={258--267},
  year={1958},
  doi={10.1063/1.1744102}
}

@article{cahn1961spinodal,
  title={On Spinodal Decomposition},
  author={Cahn, John W.},
  journal={Acta Metallurgica},
  volume={9},
  number={9},
  pages={795--801},
  year={1961},
  doi={10.1016/0001-6160(61)90182-1}
}

@article{guyer2009fipy,
  title={{FiPy}: Partial Differential Equations with {Python}},
  author={Guyer, Jonathan E. and Wheeler, Daniel and Warren, James A.},
  journal={Computing in Science \& Engineering},
  volume={11},
  number={3},
  pages={6--15},
  year={2009},
  doi={10.1109/MCSE.2009.52}
}