ajthor/burgers-fenics

1D Burgers Equation Dataset (FEniCS Implementation)

This dataset generates samples from the 1D viscous Burgers equation using FEniCS/DOLFINx for robust finite element simulation of nonlinear PDE dynamics.

Mathematical Formulation

Equation: 1D viscous Burgers equation

∂u/∂t + u∂u/∂x = ν∂²u/∂x²,   x ∈ [0, 1], t ∈ [0, T]

Boundary Conditions: Homogeneous Dirichlet

u(0, t) = u(1, t) = 0

Initial Conditions: Smooth random functions from Gaussian process

u(x, 0) = u₀(x) ~ GP(0, k(x,x'))

Physical Significance

The Burgers equation models:

• Nonlinear advection (u∂u/∂x): Wave steepening leading to shock formation
• Viscous diffusion (ν∂²u/∂x²): Smoothing and energy dissipation
• Competition between nonlinearity and diffusion at different viscosity scales

This makes it ideal for studying operator learning on nonlinear PDEs with shock dynamics.

Dataset Features

• Random initial conditions: GP sampling ensures smooth, diverse profiles
• Random viscosity: ν ∈ [0.01, 0.1] for multi-scale dynamics
• Full trajectories: Complete space-time evolution for operator learning
• Robust numerics: FEniCS + Newton method handles nonlinear terms accurately

Variables

The dataset returns a dictionary with the following fields:

Coordinates

• spatial_coordinates: (Nx,) - Physical x coordinates on scaled domain
• time_coordinates: (nt,) - Time evolution points

Solution Fields

• u_initial: (Nx,) - Initial condition u₀(x)
• u_trajectory: (nt, Nx) - Complete space-time solution u(x,t)

Parameters

• viscosity: Viscosity coefficient ν (randomly sampled per sample)
• domain_length: Physical domain length (for coordinate scaling)
• timestep: Time integration step size used
• save_interval: Trajectory sampling interval

Dataset Parameters

• Computational domain: [0, 1] (unit interval)
• Physical domain: Scaled to specified length (default: 2π)
• Default grid points: 128 spatial points
• Default time range: [0, 2.0]
• Spatial resolution: P1 Lagrange finite elements
• Temporal resolution: 0.01 time units with backward Euler

PDE-Specific Parameters

• Viscosity range: ν ∈ [0.01, 0.1] (randomly sampled)
• Boundary conditions: Homogeneous Dirichlet (u = 0 at boundaries)
• Initial conditions: GP-sampled smooth functions with zero BCs
• Time integration: Implicit backward Euler with Newton solver

Installation

pip install -r requirements.txt

Note: Requires FEniCS/DOLFINx installation. Run in appropriate container or environment.

Usage

Basic Usage

from dataset import BurgersDataset

# Create dataset
dataset = BurgersDataset(
    Lx=2*np.pi,           # Physical domain scaling
    Nx=128,               # Spatial resolution
    stop_sim_time=2.0,    # Simulation time
    timestep=0.01,        # Time step
    save_interval=10      # Trajectory sampling
)

# Generate sample
sample = next(iter(dataset))
print(f"Initial condition shape: {sample['u_initial'].shape}")
print(f"Trajectory shape: {sample['u_trajectory'].shape}")
print(f"Viscosity: {sample['viscosity']:.4f}")

Generate Dataset

python generate_data.py  # Creates train/test splits as parquet files

Visualization

Run the plotting scripts to visualize samples:

python plot_sample.py      # 3-panel plot: initial, evolution, comparison
python plot_animation.py   # Animated shock formation and evolution

Implementation Details

• Solver: DOLFINx finite elements with Newton's method
• Time integration: Implicit backward Euler for stability
• Spatial discretization: P1 Lagrange elements on unit interval
• Nonlinear solver: Newton iteration with automatic Jacobian
• Boundary conditions: Essential BCs enforced strongly

Files

• dataset.py: Main BurgersDataset class with FEniCS solver
• generate_data.py: Batch generation and parquet export
• plot_sample.py: 3-panel visualization (initial, evolution, comparison)
• plot_animation.py: Time evolution animation with shock tracking
• requirements.txt: Dependencies including FEniCS/DOLFINx

Data Generation

Generate the full dataset:

python generate_data.py

This creates train/test splits saved as chunked parquet files in the data/ directory.