Poisson 2D

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This example solves the classical 2D Poisson equation on the unit square and compares two ways of computing the Laplacian: automatic differentiation and finite differences.

Problem Setup

-Delta u(x,y) = 2 pi^2 sin(pi x) sin(pi y),   (x,y) in [0,1]^2,
u = 0 on the boundary

with exact solution

u(x,y) = sin(pi x) sin(pi y)

Step 1: Build a 2D Domain

The script creates a rectangular domain, samples interior points, and stores the exact solution and forcing on those points.

Step 2: Define a Reusable Solver Factory

The central idea is make_solver(...), which builds the same model twice while only changing the Laplacian scheme.

Step 3: Use a Hard Boundary Envelope

The neural field is multiplied by x(1-x)y(1-y), so homogeneous Dirichlet boundary conditions are satisfied by construction.

Step 4: Train Two Solvers

The script runs one solver with scheme="automatic_differentiation" and a second with scheme="finite_difference".

Step 5: Compare Predictions

After training, both predicted fields are evaluated on the same point cloud and compared against the exact solution.

What To Notice

This is a good benchmark for understanding jNO differential operator backends.
The same problem setup can be used to compare modeling choices without changing the PDE itself.
The script also contains an optional architecture-choice sweep demo using jnn.choice(...).

Download full script Back to 02 Elliptic