Shared Physics Responses Recover Hidden Rankings in Neural Operator Libraries

What the math gives to ML

The paper identifies an exact finite-dimensional reduction for label-free model selection under squared Hilbert-space loss: every pairwise preference depends only on the target's inner products with the span of candidate differences, not on the full hidden target. This suggests replacing expensive per-candidate physical diagnostics with one shared anchor-based estimate of the target projection, then ranking all experts using cheap inner products. The most promising neural-network transfer is a physics-aware deployment router for libraries of operator networks or MoE experts, where one residual/Jacobian response is computed once and reused to select or combine candidates.