Cahn-Hilliard (`cahn_hilliard`)

Spinodal decomposition: the model where the pattern is produced by the dynamics rather than supplied in the initial condition. Every sample starts from a near-uniform mixture plus small white noise, the spinodal instability amplifies a band of wavenumbers around \(\lambda^* \approx 2\pi\sqrt{2}\, \varepsilon\), and the conserved nonlinear dynamics sharpen that band into the labyrinthine or droplet morphology the equation is known for.

The conservation is the substantive difference from allen_cahn_2d. Here the spatial mean of \(u\) is preserved exactly, so mean_composition is a genuine morphology control instead of a transient.

Cahn-Hilliard — The u = 0 interface in 3D (`cahn_hilliard`): the same model runs 2D or 3D from the `resolution` dict.

Equation

\[\frac{\partial u}{\partial t} = M\,\nabla^2\!\left(u^3 - u - \varepsilon^2 \nabla^2 u\right)\]

with periodic boundary conditions, in two or three dimensions.

Operator learning task

\[u(\mathbf{x}, 0) \mapsto u(\mathbf{x}, T)\]

Parameters

Parameter	Default	Range	Description
`epsilon`	0.01	(0.004, 0.025)	Interface width; sets the pattern scale \(\lambda^* \approx 2\pi\sqrt{2}\,\varepsilon\)
`mobility`	1.0	(0.01, 10.0)	Mobility \(M\); higher separates and coarsens faster
`mean_composition`	0.0	(-0.6, 0.6)	Spatial mean of \(u\), conserved exactly
`time_end`	0.1	(0.001, 10.0)	Final time; coarsening goes as \(t^{1/3}\)
`binarize`	`False`		Return hard \(\{0, 1\}\) masks instead of the continuous field

mean_composition selects the morphology: 0 gives a bicontinuous labyrinth, and \(|m| \to 0.4\) gives minority-phase droplets. Beyond \(\pm 0.577\) the mixture leaves the spinodal regime altogether and no instability grows.

Resolution follows epsilon

The pattern scale is proportional to \(\varepsilon\), so below about \(\varepsilon = 0.006\) use a resolution of 256 or more if the interfaces are to be resolved rather than sampled.

Usage

from pdeforge import generate_dataset

# 2D labyrinth
dataset = generate_dataset(
    model="cahn_hilliard",
    n_samples=500,
    resolution={"x": 128, "y": 128},
    params={"epsilon": 0.01, "mean_composition": 0.0, "time_end": 0.1},
    seed=42,
)

# 3D: add a z axis and the same code path runs
volumes = generate_dataset(
    model="cahn_hilliard",
    n_samples=50,
    resolution={"x": 64, "y": 64, "z": 64},
    seed=4,
)

# hard two-phase masks, thresholded at u = 0
masks = generate_dataset(
    model="cahn_hilliard",
    n_samples=500,
    resolution={"x": 128, "y": 128},
    params={"binarize": True},
    seed=42,
)

The dimension is inferred from the resolution dict: {"x", "y"} runs 2D and {"x", "y", "z"} runs 3D, with no other change.

Solver

Spectral, with the fourth-order linear operator integrated exactly so the time step is not tied to \(\Delta x^4\). Randomised samples come from seeding the white-noise initial condition; the spinodal appearance is produced by the PDE.

Behaviour

Coarsening follows the Lifshitz-Slyozov law, with the characteristic domain size growing as \(t^{1/3}\). Two samples at different seeds share their morphology statistics while differing pointwise everywhere, which makes this a natural model for asking whether an operator has learned the statistics or memorised the realisations.

The binarize=True output is what porous_darcy_fem consumes: thresholding the phase field gives a two-phase medium that a flow solver can then be run across.

Data shapes

dataset.inputs.shape   # (n_samples, ny, nx)  or (n_samples, nz, ny, nx)
dataset.outputs.shape  # the same

allen_cahn_2d: the non-conserved counterpart, where domains can vanish entirely.
porous_darcy_fem: flow through the microstructure this model grows.
gray_scott_2d: the other pattern-forming system, with patterns selected by kinetics instead of composition.

Cahn-Hilliard (cahn_hilliard)