JakobEWagner/kuramoto_sivashinsky_s1024_x1ep1_t25ep1_dt5em1

The Kuramoto-Sivashinsky dataset provides numerical simulations of the Kuramoto-Sivashinsky (KS) equation, a nonlinear partial differential equation often used to model instabilities and turbulence in physical systems. The dataset is generated using a Second-Order Runge-Kutta Exponential Time Differencing Method, specifically designed for stiff systems, as discussed in Cox and Matthews' 2002 paper on exponential time differencing.

Dataset Details

• Initial Conditions: The initial conditions for the simulations are tensors of shape (batch_size, n_spatial_discretization), generated using a combination of cosine functions with random coefficients to introduce variability. The solver and equation are "warmed up" by performing 10% of the total temporal data. The final state of the warm up period is the initial condition for the dataset.
• Parameters:
  • dt: Time step (0.5 in this dataset).
  • t_end: Final time (250.0 in this dataset).
  • x_end: Length of the spatial domain (10.0 in this dataset).
  • Warmup phase: The system is warmed up by running 10% of the total time steps before starting the main simulation.
  • Train Samples: 1024 (The number of independent train samples).
  • Test Samples: 256 (The number of independent test samples).
  • n_spatial: Number of equidistant samples of the spatial domain (256 in this dataset).

Generation Method

1. Initialization: Random coefficients and cosine functions.
2. Simulation:

• Use the Second-Order Runge-Kutta Exponential Time Differencing Method to solve the KS equation over the defined spatial and temporal domains.

Usage

The dataset is saved as a tensor file (kuramoto_sivashinsky.pt) and can be loaded using PyTorch. It contains the evolution of the system over time, which can be used for various purposes such as studying chaotic behavior, training machine learning models for time series prediction, and more.

import torch

# Load the dataset
u_n = torch.load('kuramoto_sivashinsky.pt')

# u_n is a tensor of shape (batch_size, n_steps + 1, n_spatial_discretization)
print(u_n.shape)

References

Cox, Steven M., and Paul C. Matthews. "Exponential time differencing for stiff systems." Journal of Computational Physics 176.2 (2002): 430-455.