Heat 2D (`heat_2d`)

Isotropic diffusion on the square. The map is the same low-pass filter as in one dimension, now with the decay rate set by \(|\mathbf{k}|^2 = k_x^2 + k_y^2\), which means the corner modes of the spectrum die far faster than the axis modes. Structures shrink into blobs, and the blobs merge.

Equation

\[\frac{\partial u}{\partial t} = \alpha \nabla^2 u = \alpha\left(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}\right)\]

with periodic boundary conditions.

Operator learning task

\[u(x, y, 0) \mapsto u(x, y, T)\]

Parameters

Parameter	Default	Range	Description
`diffusivity`	0.01	(1e-6, 1.0)	Thermal diffusivity \(\alpha\)
`time_end`	1.0	(0.01, 10.0)	Final time \(T\)

Usage

from pdeforge import generate_dataset

dataset = generate_dataset(
    model="heat_2d",
    n_samples=1000,
    resolution={"x": 64, "y": 64},
    params={"diffusivity": 0.01, "time_end": 1.0},
    seed=42,
)

Solver

Exact spectral propagation, as in one dimension: fftn over both axes, the symbol \(e^{-\alpha |\mathbf{k}|^2 t}\) applied once, one inverse transform. A sample costs two FFTs regardless of the horizon.

Behaviour

The useful regime for a benchmark is the one where the output still carries structure the input does not trivially predict. Solving \(e^{-\alpha T |\mathbf{k}|^2} = 0.01\) at the default \(\alpha T = 0.01\) puts the 1%-of-amplitude cutoff at \(|\mathbf{k}| \approx 21\), which at \(64^2\) leaves most of the resolvable spectrum alive. Raising \(\alpha T\) to 0.1 drops that cutoff to \(|\mathbf{k}| \approx 7\), and the target starts to look like a handful of modes.

Data shapes

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

heat_1d, heat_3d: the same model in one and three dimensions.
stochastic_heat_2d: forced by space-time noise, with an ensemble per initial condition.
allen_cahn_2d: diffusion plus a double-well reaction, which stops the smoothing and holds interfaces instead.

Heat 2D (heat_2d)