SPADE: Structure-Prior Adaptive Decision Estimation

Yifan Wang

cs.AIJun 22, 2026

SPADE: Structure-Prior Adaptive Decision Estimation

Authors: Yifan Wang

Organizations: Department of Mechanical Engineering, McGill University · QC, H3A 2T7, Canada

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.

Explore similar work

Jun 15, 2026cs.LG

SciML in the Wild: A Diagnostic Study of When Structural Priors Help and When They Hurt

Scientific Machine Learning (SciML) methods such as Neural Ordinary Differential Equations (NODEs), Physics-Informed Neural Networks (PINNs), and Universal Differential Equations (UDEs) are most effective when structural priors reflect reliable governing dynamics. We ask what happens when this assumption is violated. Using macroeconomic forecasting as a stress-test domain, we evaluate five model families, ARIMA, LSTM, NODE, PINN, and UDE, across 23 countries using sparse annual data, multiple temporal splits, and five random seeds. Our results show that none of the evaluated models achieve consistently strong forecasting performance, highlighting the difficulty of low-frequency macroeconomic prediction. However, a clear relative hierarchy emerges: less-constrained models, particularly ARIMA and NODE, consistently outperform more-constrained heuristic-prior models such as PINN and UDE. Rather than treating this as a rejection of SciML, we interpret it as a diagnostic result: structural priors can act as misregularizers when they do not match the data-generating process. We identify failure modes including prior misalignment, regime shifts, structural breaks, and optimization instability, and argue that SciML practitioners should test whether structure helps before assuming that more structure is beneficial.

Vrishank Sai Anand, Prathamesh Dinesh Joshi, Raj Abhijit Dandekar +2

Prior Knowledge Machine Learning Neural Ordinary Differential Equations

Ask PDF

May 11, 2026cs.LG

The Value of Mechanistic Priors in Sequential Decision Making

Hybrid mechanistic models, physical priors with learned residuals, promise to reduce the data required for good decisions, but have no computable criterion to test this. We characterize the value of mechanistic priors in sequential decision-making within both asymptotic and burn-in regimes. To formalize this, we introduce the mechanistic information of a model -- the mutual information between the model's recommended policy

\hatπ

and the true optimal policy

π^*

-- quantified via an occupancy-weighted bias

B_μ

. In the asymptotic regime (large

N

), matched bounds reveal that Bayesian regret scales with the residual entropy

H_{\mathrm{mech}}

, delivering a theoretical sample complexity reduction of

H(μ)/H_{\mathrm{mech}}

compared to an uninformed baseline. Furthermore, we provide a model certificate to determine empirical sample efficiency. Complementarily, in the clinically relevant burn-in regime (small

N

), we establish a lower bound on the penalty incurred by confidently wrong priors. We demonstrate both the asymptotic and burn-in bounds across 5-fluorouracil (5-FU) dosing simulations motivated by published FOLFOX pharmacokinetic data, where a hybrid prior yields large sample-efficiency gains in the burn-in regime. Finally, we contrast these grounded models with LLM priors, demonstrating that LLMs can suffer severe losses in mechanistic information, thereby motivating the exclusive use of physically-grounded priors for safety-critical applications.

Itai Shufaro, Gal Benor, Shie Mannor

Sequential Decision Making Prior Knowledge Linear Regret Decisions Drug-Drug Interactions Value

Ask PDF

May 29, 2026cs.LG

Scalable Bayesian Inference for Nonlinear Conservation Laws

Nonlinear conservation laws are at the heart of many of the most important dynamical systems in science and engineering. In practical applications, such systems are often subject to various sources of uncertainty, e.g. due to sparse or noisy measurements. Inferring physical quantities and fields of interest then becomes an ill-posed problem which both classical numerical methods and modern deep learning-based methods struggle to treat appropriately. Recent work has framed classical numerical methods as Bayesian inference under Gaussian process priors, resulting in a physics-aware treatment of uncertainties. Following this line of work, we develop a novel numerically conservative method for uncertainty-aware simulations of nonlinear conservation laws. We use recent sparse approximation techniques to scale up to large-scale forward and inverse problems. For forward simulation, we inherit the accuracy of classical solvers while providing structured uncertainty quantification. On inverse problems, we recover posteriors over nonparametric source fields in seconds -- outperforming neural baselines that take minutes to produce a less accurate point estimate.

Tim Weiland, Philipp Hennig

Reliable Uncertainty Quantification Ill-Posed Inverse Problem Bayesian Classical Numerical Solver Gaussian Process

Ask PDF