metadata

language:
  - en
license: cc-by-4.0
tags:
  - physics
task_categories:
  - time-series-forecasting
  - other
task_ids:
  - multivariate-time-series-forecasting

This Dataset is part of The Well Collection.

How To Load from HuggingFace Hub

Be sure to have the_well installed (pip install the_well)
Use the WellDataModule to retrieve data as follows:

from the_well.data import WellDataModule

# The following line may take a couple of minutes to instantiate the datamodule
datamodule = WellDataModule(
    "hf://datasets/polymathic-ai/",
    "gray_scott_reaction_diffusion",
)
train_dataloader = datamodule.train_dataloader()

for batch in dataloader:
    # Process training batch
    ...

Pattern formation in the Gray-Scott reaction-diffusion equations

One line description of the data: Stable Turing patterns emerge from randomness, with drastic qualitative differences in pattern dynamics depending on the equation parameters.

Longer description of the data: The Gray-Scott equations are a set of coupled reaction-diffusion equations describing two chemical species, $A$ and $B$ , whose concentrations vary in space and time. The two parameters $f$ and $k$ control the “feed” and “kill” rates in the reaction. A zoo of qualitatively different static and dynamic patterns in the solutions are possible depending on these two parameters. There is a rich landscape of pattern formation hidden in these equations.

Associated paper: None.

Domain expert: Daniel Fortunato, CCM and CCB, Flatiron Institute.

Code or software used to generate the data: Github repository (MATLAB R2023a, using the stiff PDE integrator implemented in Chebfun. The Fourier spectral method is used in space (with nonlinear terms evaluated pseudospectrally), and the exponential time-differencing fourth-order Runge-Kutta scheme (ETDRK4) is used in time.)

Equation describing the data

$\begin{align*} \frac{\partial A}{\partial t} &= \delta_A\Delta A - AB^2 + f(1-A) \\ \frac{\partial B}{\partial t} &= \delta_B\Delta B - AB^2 - (f+k)B \end{align*}$

The dimensionless parameters describing the behavior are: $f$ the rate at which $A$ is replenished (feed rate), $k$ the rate at which $B$ is removed from the system, and $\delta_A, \delta_B$ the diffusion coefficients of both species.

Dataset	FNO	TFNO	Unet	CNextU-net
`gray_scott_reaction_diffusion`	$\mathbf{0.1365}$	0.3633	0.2252	$\mathbf{0.1761}$

Table: VRMSE metrics on test sets (lower is better). Best results are shown in bold. VRMSE is scaled such that predicting the mean value of the target field results in a score of 1.

About the data

Dimension of discretized data: 1001 time-steps of 128 $\times$ 128 images.

Fields available in the data: The concentration of two chemical species $A$ and $B$ .

Number of trajectories: 6 sets of parameters, 200 initial conditions per set = 1200.

Estimated size of the ensemble of all simulations: 153.8 GB.

Grid type: uniform, cartesian coordinates.

Initial conditions: Two types of initial conditions generated: random Fourier series and random clusters of Gaussians.

Boundary conditions: periodic.

Simulation time-step: 1 second.

Data are stored separated by ( $\Delta t$ ): 10 seconds.

Total time range ( $t_{min}$ to $t_{max}$ ): $t_{min} =0$ , $t_{max} = 10,000$ .

Spatial domain size ( $L_{x}$ , $L_{y}$ ): $[-1,1]\times[-1,1]$ .

Set of coefficients or non-dimensional parameters evaluated: All simulations used $\delta_u = 2.10^{-5}$ and $\delta_v = 1.10^{-5}$ . "Gliders": $f = 0.014, k = 0.054$ . "Bubbles": $f = 0.098, k = 0.057$ . "Maze": $f = 0.029, k = 0.057$ . "Worms": $f = 0.058, k = 0.065$ . "Spirals": $f = 0.018, k = 0.051$ . "Spots": $f = 0.03, k = 0.062$ .

Approximate time to generate the data: 5.5 hours per set of parameters, 33 hours total.

Hardware used to generate the data: 40 CPU cores.

What is interesting and challenging about the data:

What phenomena of physical interest are catpured in the data: Pattern formation: by sweeping the two parameters $f$ and $k$ , a multitude of steady and dynamic patterns can form from random initial conditions.

How to evaluate a new simulator operating in this space: It would be impressive if a simulator—trained only on some of the patterns produced by a subset of the $(f, k)$ parameter space—could perform well on an unseen set of parameter values $(f, k)$ that produce fundamentally different patterns. Stability for steady-state patterns over long rollout times would also be impressive.

Warning: Due to the nature of the problem and the possibility to reach an equilibrium for certain values of the kill and feed parameters, a constant stationary behavior can be reached. Here are the trajectories for which a stationary behavior was identified for specy $A$ as well as the corresponding time at which it was reached:

Validation set:
- $f = 0.014, k = 0.054$ $f = 0.014, k = 0.054$ :
  - Trajectory 7, time = 123
  - Trajectory 8, time = 125
  - Trajectory 10, time = 123
  - Trajectory 11, time = 125
  - Trajectory 12, time = 121
  - Trajectory 14, time = 121
  - Trajectory 15, time = 129
  - Trajectory 16, time = 124
  - Trajectory 17, time = 122
  - Trajectory 18, time = 121
  - Trajectory 19, time = 155
- $f = 0.018, k = 0.051$ $f = 0.018, k = 0.051$ :
  - Trajectory 14, time = 109
Training set:
- $f = 0.014, k = 0.054$ $f = 0.014, k = 0.054$ :
  - Trajectory 81, time = 126
  - Trajectory 82, time = 126
  - Trajectory 83, time = 123
  - Trajectory 85, time = 123
  - Trajectory 86, time = 124
  - Trajectory 87, time = 127
  - Trajectory 88, time = 121
  - Trajectory 90, time = 123
  - Trajectory 91, time = 121
  - Trajectory 92, time = 126
  - Trajectory 93, time = 121
  - Trajectory 94, time = 126
  - Trajectory 95, time = 125
  - Trajectory 96, time = 123
  - Trajectory 97, time = 126
  - Trajectory 98, time = 121
  - Trajectory 99, time = 125
  - Trajectory 100, time = 126
  - Trajectory 101, time = 125
  - Trajectory 102, time = 159
  - Trajectory 103, time = 129
  - Trajectory 105, time = 125
  - Trajectory 107, time = 122
  - Trajectory 108, time = 126
  - Trajectory 110, time = 127
  - Trajectory 111, time = 122
  - Trajectory 112, time = 121
  - Trajectory 113, time = 122
  - Trajectory 114, time = 126
  - Trajectory 115, time = 126
  - Trajectory 116, time = 126
  - Trajectory 117, time = 122
  - Trajectory 118, time = 123
  - Trajectory 119, time = 123
  - Trajectory 120, time = 125
  - Trajectory 121, time = 126
  - Trajectory 122, time = 121
  - Trajectory 123, time = 122
  - Trajectory 125, time = 125
  - Trajectory 126, time = 127
  - Trajectory 127, time = 125
  - Trajectory 129, time = 125
  - Trajectory 130, time = 122
  - Trajectory 131, time = 125
  - Trajectory 132, time = 131
  - Trajectory 133, time = 126
  - Trajectory 134, time = 159
  - Trajectory 135, time = 121
  - Trajectory 136, time = 126
  - Trajectory 137, time = 125
  - Trajectory 138, time = 126
  - Trajectory 139, time = 123
  - Trajectory 140, time = 128
  - Trajectory 141, time = 126
  - Trajectory 142, time = 123
  - Trajectory 144, time = 122
  - Trajectory 145, time = 125
  - Trajectory 146, time = 123
  - Trajectory 147, time = 126
  - Trajectory 148, time = 121
  - Trajectory 149, time = 122
  - Trajectory 150, time = 125
  - Trajectory 151, time = 126
  - Trajectory 152, time = 152
  - Trajectory 153, time = 127
  - Trajectory 154, time = 122
  - Trajectory 155, time = 124
  - Trajectory 156, time = 122
  - Trajectory 158, time = 126
  - Trajectory 159, time = 121
- $f = 0.018, k = 0.051$ $f = 0.018, k = 0.051$ :
  - Trajectory 97, time = 109
  - Trajectory 134, time = 107
  - Trajectory 147, time = 109
  - Trajectory 153, time = 112
Test set:
- $f = 0.014, k = 0.054$ $f = 0.014, k = 0.054$ :
  - Trajectory 12, time = 127
  - Trajectory 13, time = 125
  - Trajectory 14, time = 123
  - Trajectory 15, time = 126
  - Trajectory 16, time = 126
  - Trajectory 17, time = 123
  - Trajectory 18, time = 128
  - Trajectory 19, time = 125
- $f = 0.018, k = 0.051$ $f = 0.018, k = 0.051$ :
  - Trajectory 11, time = 113