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

Neural networks trained under different hyperparameter settings can fall into distinct training "regimes," with consistent behavior within regimes and qualitative differences across regimes. In this paper, we study such multi-regime behavior in scientific machine learning (SciML) models through a regime-aware diagnostic framework that jointly analyzes performance, training dynamics, and loss-landscape geometry. We identify three key findings: (i) a consistent three-regime structure emerges across many standard SciML models, different constraint enforcements, and various optimizer designs; (ii) optimization effectiveness is regime-specific, with no single method performing well across all regimes; and (iii) SciML models can exhibit fine-grained failure modes that can challenge conventional interpretations of standard loss-landscape metrics. Our results provide an approach to establish a unified, task-oblivious perspective on failure modes in SciML and to inform regime-aware guidance for improving robustness. We validate these findings across widely-used SciML models, including physics-informed neural networks, neural operators, and neural ordinary differential equations, on benchmarks spanning representative ordinary and partial differential equations.

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.

We introduce a neural network-based framework for learning time series estimators through a process we term decision-theoretic pretraining. Analysts specify a generative world, a distribution over data-generating processes, and a target decision objective. A neural network trained on stratified simulations from this world approximates the corresponding optimal decision rule, yielding a neural estimator that provides forecasts, parameter estimates, predictive intervals, or model-selection for zero-shot inference on previously unseen time series. The joint specification of the generative world and objective enables the estimators to directly approximate process-level, finite-sample properties: near-optimal risk, bias control, minimax performance, and uniform calibration. Our experiments demonstrate that these neural estimators can outperform traditional baselines such as maximum likelihood estimation and model selection via AICc, for the same model structural model classes. Furthermore, even when trained purely on simulations of structural models, they achieve competitive or state-of-the-art forecasting accuracy on major real-world benchmarks, compared with statistical, neural or large pre-trained models. We illustrate the framework by addressing two longstanding challenges: finite-sample bias and miscalibration in AR(p) models, and the forecast combination puzzle. These applications highlight the approach's main advantage: its ability to approximate solutions to analytically intractable or computationally prohibitive time series problems, including complex structural equations or optimality criteria. Ultimately, by enabling explicit control over decision-theoretic trade-offs, the framework equips analysts with highly efficient estimation tools tailored to their specific analytical needs.