Ready-wired steady heat demo¶

Load with File → Examples → Heat demo. The separate basic template remains available through Reset Poisson template.

Problem and configuration¶

The baseline equation is Δu + c = 0 on the unit square, with c=2 and u=0 on every edge. Here u is dimensionless temperature rise, conductivity is one, and the source is constant. This is a steady problem, not a transient heat-equation solver.

The demo uses the existing methods/pinn.py recipe, configured in templates/heat.py: 144 grid interior points, 24 boundary samples per side, a tanh network with hidden layers 32,32, Adam learning rate .005 and 2500 updates. The PDE weight is 1 and mean BC weight is 10. Geometry, conditions and training settings remain editable in their blocks. Boundary conditions are imposed softly by losses; finite training does not imply exact satisfaction or a prescribed accuracy.

Output wiring¶

PINN method ──→ Training Monitor
            ├─→ Solution Plot
            └─→ Comparison ←── Reference

The monitor records all 2500 default updates within its 10000-update retention limit. Views open only by explicit block action. The prediction block has no implicit reference. Comparison requests its own evaluation points and presents prediction, reference and signed error fields, plus RMSE and max |error|. Prediction/reference maps share a colour scale; the error map has its own range.

Rectangular reference¶

For rectangle widths Lx,Ly and a common constant Dirichlet value B, the reference subtracts B and expands the constant forcing in odd double-sine modes. Let ξ=x-x_min and η=y-y_min. The infinite-series coefficients are

A_mn = 16 c / [π² m n ((mπ/Lx)² + (nπ/Ly)²)],  m,n odd
u = B + Σ A_mn sin(mπξ/Lx) sin(nπη/Ly).

The implementation retains 40 odd modes per axis (indices 1,3,...,79), evaluates in float64 chunks, and labels the source as truncated. It is a numerical reference from an analytical expansion, not an exact finite formula or universal error bound.

The separation-of-variables solution formula is documented in University of Maryland: rectangular Dirichlet Poisson equation. The constant-source odd coefficients above are our specialization of that formula; no solver code from the page was copied.

Supported rectangle references require Dirichlet conditions on every side with the same constant target. Mixed/Neumann rectangle references are not provided; remove the Comparison if using a problem with no supported reference. Interval comparisons retain their existing analytical reference.

tests/test_rectangle_reference.py checks zero boundaries, symmetry and selected 40-versus-80-mode values, and compares against an independent five-point finite- difference solve on a translated, non-square rectangle with a nonzero boundary. These are correctness checks at stated resolutions, not a rigorous truncation bound. tests/test_heat_demo_ui.py verifies the wired demo, background run, monitor history, reference metrics, comparison rendering and return to the basic template.

Measured soft/hard BC performance is recorded in the accuracy study and results catalogue. These measurements are separate from the reference correctness checks and are not general accuracy guarantees.