SPADE: Structure-Prior Adaptive Decision Estimation
2026-06-22 • Artificial Intelligence
Artificial Intelligence
AI summaryⓘ
The authors present SPADE, a new method that helps machines decide when and how to use known physical laws, like conservation or symmetry, to make predictions. Instead of always assuming these laws apply, SPADE tests if data actually supports them and adjusts how strongly to apply them accordingly. This approach improves prediction accuracy, selects the correct physical laws reliably, and does so more efficiently than current methods. The method works well across different types of physical assumptions and outperforms some neural network baselines.
physical priorsconservation lawsHamiltonian systemsStein shrinkagestructure selectionJames-Stein estimatorstructure priorBenjamini-Hochberg procedurecross-validationscientific machine learning
Authors
Yifan Wang
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.