Darcy 2D (`darcy_2d`)

Steady flow through a porous medium, on the periodic box with a spectral solver. The task is the classic one: hand the operator a permeability field and ask for the pressure it induces. Where darcy_fno_2d reproduces the published benchmark down to its grid convention, darcy_2d is the freer periodic version, useful when you want the physics without inheriting somebody else's measure.

Equation

\[-\nabla \cdot \big(\kappa(x,y)\,\nabla u\big) = f\]

with periodic boundary conditions.

Operator learning task

\[\kappa(x, y) \mapsto u(x, y)\]

Parameters

Parameter	Default	Range	Description
`kappa_min`	0.1	(1e-3, 10.0)	Lower end of the permeability range
`kappa_max`	10.0	(1.0, 100.0)	Upper end of the permeability range
`source_type`	`"sine"`	`"sine"`, `"constant"`	Forcing \(f\)

The contrast ratio \(\kappa_{\max}/\kappa_{\min}\) is the parameter that actually sets the difficulty. At the defaults it is 100, which already puts sharp gradients at the interfaces.

describe_model will not list these

They are read from the model's defaults dictionary rather than declared as tunable specs, so describe_model("darcy_2d") prints Parameters: See DEFAULT_PARAMS instead of the table above. Passing them through params= works as documented.

Usage

from pdeforge import generate_dataset

dataset = generate_dataset(
    model="darcy_2d",
    n_samples=1000,
    resolution={"x": 64, "y": 64},
    params={"kappa_min": 0.1, "kappa_max": 10.0},
    seed=42,
)

Solver

Spectral discretisation with conjugate-gradient iteration for the variable-coefficient operator. The periodic setting is what makes the FFT usable here; it is also what distinguishes this model from the Dirichlet benchmark family.

The input measure

Permeability fields come from a Gaussian random field pushed through a sigmoid, which maps the unbounded Gaussian onto \([\kappa_{\min}, \kappa_{\max}]\) and keeps the coefficient strictly positive so the operator stays elliptic. The result reads as smoothly channelised media rather than the two-phase blocks of the piecewise-constant benchmark.

Behaviour

High contrast puts steep pressure gradients along the permeability interfaces, and those interfaces are where a band-limited operator loses accuracy first; smooth permeability gives smooth pressure and a correspondingly easy problem. Wherever the sigmoid produces channelised structures the flow concentrates into them, so a small area of the domain carries most of the signal and an averaged error metric will under-report what went wrong.

Data shapes

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

darcy_fno_2d: the canonical benchmark, bit-exact against the distributed data.
porous_darcy_fem: Darcy flow across a Cahn-Hilliard microstructure, solved by finite elements.
helmholtz_2d: the other steady spectral problem.

Darcy 2D (darcy_2d)