Gray-Scott 2D (`gray_scott_2d`)

The reaction-diffusion system with a parameter plane worth exploring. Pearson (1993) mapped feed rate \(F\) against kill rate \(k\) and found that perturbations of the trivial state \((U, V) = (1, 0)\) grow into spots, stripes, labyrinths or self-replicating patterns depending on where in that plane you sit. For operator learning this is unusual and useful: two nearby parameter values can give qualitatively different fields, so the map from parameters to solution is not smooth in any helpful sense.

Equation

\[\frac{\partial U}{\partial t} = D_U \nabla^2 U - U V^2 + F\,(1 - U)\]

\[\frac{\partial V}{\partial t} = D_V \nabla^2 V + U V^2 - (F + k)\,V\]

on the periodic box, in Pearson's scaling: domain \([0, 2.5]^2\), \(D_U = 2\times10^{-5}\), \(D_V = 10^{-5}\).

Operator learning task

\[(U, V)(x, y, 0) \mapsto (U, V)(x, y, T)\]

with the two fields stacked on a leading component axis, shape (2, ny, nx).

Parameters

Parameter	Default	Range	Description
`feed`	0.04	(0.0, 0.12)	Feed rate \(F\)
`kill`	0.06	(0.0, 0.08)	Kill rate \(k\)
`Du`	2e-5	(1e-6, 1e-3)	Diffusivity of \(U\)
`Dv`	1e-5	(1e-6, 1e-3)	Diffusivity of \(V\)
`time_end`	2000.0	(10.0, 20000.0)	Final time; patterns need \(t \sim 1000\) or more

Usage

from pdeforge import generate_dataset

dataset = generate_dataset(
    model="gray_scott_2d",
    n_samples=200,
    resolution={"x": 128, "y": 128},
    params={"feed": 0.054, "kill": 0.063, "time_end": 8000.0},
    backend="jax",
    seed=11,
)

A few reference points in the Pearson plane: \((0.054, 0.063)\) gives coral-like growth, \((0.0367, 0.0649)\) gives a space-filling labyrinth, and \((0.04, 0.06)\), the default, gives mitosis-like self-replicating spots.

Solver

The two components ride the spectral seam's leading component axis with a diagonal linear symbol, and the \(UV^2\) kinetics is stepped explicitly.

Behaviour

Patterns take time. Below \(t \approx 1000\) the field is still a perturbation of the trivial state, and a dataset generated at short horizon will mostly record the initial seeding rather than the pattern. The default time_end of 2000 is the lower end of the useful range.

The initial condition matters more than usual, because the pattern grows out of where the seeds were placed. The n_patches parameter of the generator sets how many seeds start the growth, and a single seed on a large domain gives a very different picture from forty.

Data shapes

dataset.inputs.shape   # (n_samples, 2, ny, nx)
dataset.outputs.shape  # (n_samples, 2, ny, nx)

fitzhugh_nagumo_2d: the other two-component excitable system, with travelling fronts instead of stationary patterns.
cahn_hilliard: pattern formation driven by a conservation law rather than by kinetics.
lotka_volterra_2d: predator-prey kinetics with diffusion.

Gray-Scott 2D (gray_scott_2d)