SPADE: Structure-Prior Adaptive Decision Estimation

Abstract

Physical-structure priors such as conservation laws, Hamiltonian forms, and symmetries can improve scientific machine learning when correct, but can degrade predictions when misspecified. Existing methods usually enforce a chosen structure or tune a soft penalty, without a calibrated rule for deciding whether to impose a prior, how strongly to impose it, which prior to use, or which subset of candidate laws holds. We introduce SPADE, Structure-Prior Adaptive Decision Estimation, a closed-form framework that treats this problem as shrinkage of the structure-violating block of an unconstrained estimator. SPADE uses one exact specification test and one estimand: the test decides whether the prior is supported by data, Stein-unbiased James-Stein shrinkage sets the enforcement strength with an $O(σ^2/n)$ oracle guarantee, and a gate commits to the hard prior only when the test certifies it. The same test yields consistent nested structure selection and Benjamini-Hochberg control for subset discovery in non-nested constraint families. Across a linear-subspace prior, a reservoir conservation law, and a nonlinear Hamiltonian prior on Duffing dynamics, SPADE tracks the oracle, beats a neural-network baseline, reduces correct-prior regret from $10.3\%$ to $2.6\%$, matches cross-validation with $1/71$ of the solves, selects the correct structure with $100\%$ accuracy, and recovers partial laws with controlled false relaxation.