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Stochastic Burgers 1D (stochastic_burgers_1d)

Noise on top of nonlinear dynamics, which is where uncertainty stops being tractable analytically. Burgers concentrates its structure into thin fronts, and the noise perturbs where those fronts sit, so the ensemble spread is not a smooth field: it is a set of narrow, tall ridges at the front locations. A predicted uncertainty that is smooth in space will be wrong in exactly the places that matter.

Stochastic Burgers 1D

One initial condition and five realisations of the same solve (stochastic_burgers_1d): the members agree away from the fronts and disagree at them.

Equation

\[du = \left[-u\,\frac{\partial u}{\partial x} + \nu\,\frac{\partial^2 u}{\partial x^2}\right] dt + \sigma\,dW(t, x)\]

with additive space-time noise, white in time and optionally smoothed in space, on the periodic domain.

Operator learning task

\[u_0 \mapsto \{u_T^{(1)}, u_T^{(2)}, \dots\}\]

the natural target for distributional operator learning of \(P(u_T \mid u_0)\).

Parameters

Parameter Default Range Description
viscosity 0.05 (1e-4, 1.0) Viscosity \(\nu\)
noise_intensity 0.1 (0.0, 2.0) Noise amplitude \(\sigma\)
n_realizations 10 (1, 1000) Realisations per initial condition
time_horizon 0.5 (0.01, 10.0) Final time \(T\)

Usage

from pdeforge import generate_dataset

dataset = generate_dataset(
    model="stochastic_burgers_1d",
    n_samples=200,
    resolution={"x": 256},
    params={"viscosity": 0.05, "noise_intensity": 0.1, "n_realizations": 20},
    seed=42,
)
dataset.outputs.shape   # (200, 20, 256)

Solver

Exponential integrator for the exact viscous part, explicit dealiased advection, and Euler-Maruyama noise increments: the same conventions as the stochastic heat models. Sending \(\sigma \to 0\) recovers the deterministic dynamics, which the test suite checks.

Behaviour

The interaction between noise and nonlinearity is the whole point. Because front position depends on the whole history of the field, a small perturbation early produces a displaced front later, and displacing a steep front produces a large pointwise difference. The result is that ensemble variance is concentrated where the deterministic solution has its steepest gradients, and the distribution there is skewed rather than Gaussian.

Raising viscosity widens the fronts and makes the spread better behaved; lowering it sharpens both the fronts and the failure of any Gaussian approximation.

Data shapes

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