README.md

2D Poisson Equation Dataset

Numerical solutions to the 2D Poisson equation with mixed boundary conditions using Dedalus spectral methods.

Equation

The 2D Poisson equation boundary value problem:

PDE: ∇²u = f(x,y) in Ω = [0, Lx] × [0, Ly]

Boundary Conditions:

• u(x,0) = g(x) (Dirichlet on bottom)
• ∂u/∂y(x,Ly) = h(x) (Neumann on top)

Variables

The dataset returns a dictionary with the following fields:

Coordinates

• spatial_coordinates: (2, Nx, Ny) - Combined X,Y coordinate meshgrids

Solution Fields

• solution_field: (Nx, Ny) - Solution u(x,y)
• forcing_function: (Nx, Ny) - Random forcing function f(x,y)

Boundary Conditions

• boundary_condition_bottom: (Nx,) - Bottom Dirichlet BC g(x)
• boundary_condition_top_gradient: (Nx,) - Top Neumann BC h(x)

Dataset Parameters

• Domain: [0, 2π] × [0, π] (2D rectangular domain)
• Grid points: 256 × 128 (Nx × Ny)
• Discretization: Fourier(x) × Chebyshev(y) spectral methods
• Solver: Dedalus LBVP (Linear Boundary Value Problem)

Randomization

• Forcing function: Generated using Gaussian processes with random length scales
• Boundary conditions: Fixed sinusoidal bottom BC, zero top gradient BC
• Amplitude: Random amplitude scaling for forcing functions (0.5 to 3.0)

Physical Context

This dataset simulates steady-state physical systems governed by the 2D Poisson equation. The equation models phenomena where the spatial distribution depends on source/sink terms, including:

Applications:

• Electrostatic potential in the presence of charge distributions
• Steady-state heat conduction with internal heat sources
• Fluid stream functions for incompressible flow
• Gravitational potential from mass distributions

Usage

from dataset import PoissonDataset

# Create dataset
dataset = PoissonDataset()

# Generate a sample
sample = next(iter(dataset))

# Access solution data
spatial_coords = sample["spatial_coordinates"]  # X, Y meshgrids
solution = sample["solution_field"]            # u(x,y)
forcing = sample["forcing_function"]           # f(x,y)

Visualization

Run the plotting script to visualize samples:

python plot_sample.py      # 2D visualization of forcing, solution, and BCs

Data Generation

Generate the full dataset:

python generate_data.py

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