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name: ntk_block_spectrum description: Compute PINN neural tangent kernel block traces, eigenvalue summaries, and dominance diagnostics from empirical Jacobians.

NTK Block Spectrum

Use this skill when a PINN recovery task needs to diagnose convergence-rate imbalance between boundary and residual loss components. It implements the paper's block-kernel view with small deterministic arrays.

Inputs

• Boundary Jacobian J_u as rows over boundary observations and columns over parameters.
• Residual Jacobian J_r as rows over residual observations and columns over the same parameters.
• Optional dominance tolerance.

Outputs

• trace_kuu, trace_krr, and trace_full.
• Eigenvalue lists for Kuu and Krr.
• Dominance label: residual_dominates, boundary_dominates, or balanced.

Workflow

1. Validate that both Jacobians are non-empty matrices with equal parameter dimension.
2. Compute Gram blocks Kuu = J_u J_u^T and Krr = J_r J_r^T.
3. Compute traces and symmetric eigenvalues.
4. Compare block traces and record which loss component is expected to converge faster.
5. Pass the diagnostics to the adaptive weighting skill.

Validation

Run:

python tests/test_ntk_spectrum.py

The tests use residual-dominant Jacobians and check trace, eigenvalue, and label behavior.

Limitations

This skill computes empirical finite-width diagnostics. It does not prove infinite-width convergence; it preserves the operational recovery mechanism needed for bounded experiments.