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name: pinn_residual_problem description: Build deterministic PINN residual problem fixtures with boundary and PDE residual targets for NTK recovery experiments.

PINN Residual Problem

Use this skill when a recovery task needs an explicit physics-informed neural network problem item rather than a generic regression dataset. It is appropriate for small Poisson-style probes, source-boundary checks, and module tests that need separated boundary and residual loss contracts.

Do not use this skill to claim a full paper benchmark. The default fixture is a declared synthetic proxy designed to exercise the paper mechanism.

Inputs

• frequency: positive integer or float for u(x)=sin(frequency*pi*x).
• residual_points: interior collocation points in [0, 1].
• Optional boundary points, defaulting to [0, 1].

Outputs

• A JSON-compatible problem dictionary with boundary_points, boundary_targets, residual_points, residual_targets, and operator.
• Residual targets follow the Poisson second-derivative convention u_xx(x) = -(frequency*pi)^2 sin(frequency*pi*x).

Workflow

1. Choose a deterministic set of boundary and residual points.
2. Compute exact solution values and second-derivative forcing values.
3. Validate that arrays are non-empty, finite, and aligned.
4. Pass the problem dictionary to NTK diagnostics or proxy training.

Validation

Run:

python tests/test_poisson_problem.py

The tests verify endpoint boundary values, residual sign, finite values, and metadata needed by downstream recovery.

Limitations

The fixture is intentionally small and synthetic. It preserves the paper's residual-plus-boundary loss mechanism but does not reproduce full-width neural network training or all numerical experiments.