TortillaChip/burgers1d-periodic

1D Burgers' Equation Data

Input–output pairs for the 1D viscous Burgers' equation on a periodic domain, intended for training and benchmarking neural operators.

Each sample maps an initial condition u_0(x) = u(x, 0) to the solution u(x, t_end). Initial conditions are drawn from a Gaussian random field ande the PDE is solved numerically with a pseudo-spectral method.

The 1D viscous Burgers' equation on a periodic domain $x \in [0, 1)$ reads

$$\frac{\mathrm{\partial }u}{\mathrm{\partial }t}=-u\frac{\mathrm{\partial }u}{\mathrm{\partial }x}+\nu \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2},\backslash frac\{\backslash partial\; u\}\{\backslash partial\; t\}\; =\; -u\backslash frac\{\backslash partial\; u\}\{\backslash partial\; x\}\; +\; \backslash nu\backslash frac\{\backslash partial^2\; u\}\{\backslash partial\; x^2\},$$

where $\nu > 0$ is the kinematic viscosity. The first term is a non-linear advective term and the second is a diffusive term.

Initial conditions are sampled according to

$$u_0\sim \mathcal{N}(0,625(-\mathrm{\Delta }+25I{)}^{-2}),u\_0\; \backslash sim\; \backslash mathcal\{N\}\backslash left(0,\; 625(-\backslash Delta\; +\; 25I)^\{-2\}\backslash right),$$

where $\Delta$ is the Laplacian.

To integrate the equation forward in time, the convective term is advanced with a fourth-order Runge-Kutta (RK4) method, while the diffusive term is discretised with a backward Euler method and the resulting linear system is solved exactly by a point-wise multiplication in spectral space.

Parameters

Parameter	Value
Kinematic viscosity	0.02
Spatial resolution	8192
Domain	[0, 1) (periodic)
End time	1.0
dtype	float32
Time-step size	1e-5
Number of samples	1280

Schema

Each row is one sample:

Column	Type	Description
sample_id	int64	Sample index
x	list<float32> (length 8192)	Spatial grid coordinates
u0	list<float32> (length 8192)	Initial condition
u_end	list<float32> (length 8192)	Solution

Dataset attributes (nu, dt, t_end, resolution, seed, num_samples, dtype) are stored in the accompanying metadata.json.

Details

The norax library contains the code used generate this dataset and demonstrates how it can be used to train a Fourier neural operator.

Citations

This dataset is generated according to the method described in Li, Zongyi, et al. "Fourier neural operator for parametric partial differential equations." arXiv preprint arXiv:2010.08895 (2020).