parameter interpretation

This study addresses the challenge of jointly modeling calibration and control parameters in computer model calibration, where the distribution of calibration parameters is unknown while that of control parameters is known. To tackle this issue, the authors propose a nonparametric Bayesian calibration method based on measure decomposition. The approach preserves the known marginal distribution of the control parameters while employing stochastic process modeling and Bayesian inference to construct a posterior distribution over the input space that aligns with field observations. Notably, this work is the first within a nonparametric calibration framework to explicitly maintain the prior distributional properties of the control parameters, thereby substantially enhancing the physical consistency and scientific credibility of the calibration results.

Machine learning training is significantly more time-consuming than inference, and the design of input or parameter perturbations has long relied on empirical trial-and-error. Method: This paper models training dynamics as a first-passage process and introduces a statistical mechanics framework to analyze model responses to input/parameter perturbations. It proposes, for the first time, a single-frequency perturbation response theory grounded in the quasi-stationary assumption, and rigorously proves its generalizability to multi-frequency perturbation regimes—enabling rational optimization of perturbation protocols. Contribution/Results: Evaluated on ResNet-18 trained for CIFAR-10 classification, the method precisely identifies the optimal perturbation type and frequency, reducing training iterations by 23% and improving test accuracy by 1.4 percentage points, thereby substantially enhancing both training efficiency and generalization performance.

Existing surrogate models primarily focus on identifying a single optimal parameter set, neglecting the broader distribution of parameters that satisfy a given target output. Method: We propose a joint input-output space density estimation framework that integrates neural surrogate modeling, feature likelihood estimation, and Bayesian inference to construct a confidence-aware parameter prior. This enables efficient sampling and visualization of plausible parameter sets in high-dimensional spaces. Contribution/Results: Our key innovation lies in unifying density estimation with inverse inference to support interactive exploration of multi-solution parameter distributions. Evaluated on three scientific simulation datasets, the method demonstrates effectiveness in goal-directed parameter analysis, significantly enhancing users’ understanding of and ability to control the parameter-feature mapping relationship.

Model-form uncertainty (MFU)—arising from simplifying modeling assumptions and particularly challenging to quantify during extrapolation—remains a critical, yet poorly addressed, source of epistemic uncertainty in physics-based modeling. Existing approaches heavily rely on calibration data and cannot isolate the independent influence of individual assumptions on predictions. Method: We propose a calibration-free MFU quantification framework that parameterizes modeling assumptions and integrates grouped variance-based sensitivity analysis to explicitly characterize how assumption changes propagate into predictive variance. The method accommodates parameter dependencies and enables assumption importance ranking under extrapolative conditions. Contribution/Results: Experiments demonstrate that our approach effectively identifies the assumptions dominating prediction uncertainty. It provides quantitative guidance for model simplification, verification, and refinement, thereby significantly enhancing the credibility and robustness of complex physics-based models.

Existing mathematical modeling lacks a rigorous, unambiguous ontological foundation, hindering a unified characterization of the mapping between models and real-world phenomena. This paper introduces, for the first time, an axiomatic definition of mathematical models grounded in Hilbert-space operator theory: a model is formalized as a computable operator acting on random variables, systematically unifying theoretical derivation, experimental implementation, and statistical identification. We further establish a geometric correspondence between the model manifold and the prediction surface, exposing intrinsic structural properties and the fundamental nature of model computability. This framework fills a critical gap in the formal ontology of modeling, providing a unified mathematical foundation for interdisciplinary model construction. It significantly enhances the logical rigor of theoretical inference and the reliability of empirical validation.

This study addresses the common issue in ensemble forecasting wherein insufficiently rapid growth of ensemble spread leads to inadequate representation of uncertainty. Using the Lorenz '96 system, the work systematically disentangles intrinsic variability, initial condition perturbations, and stochastic model uncertainty to evaluate how various ensemble configurations and parameterization schemes influence spread evolution. It introduces novel Bayesian and streaming stochastic parameterizations featuring temporally coherent structures, revealing that perturbations primarily govern the rate of trajectory decorrelation rather than long-term variance. The analysis further elucidates the interaction mechanisms among distinct uncertainty sources. Experimental results demonstrate that the proposed methods significantly enhance early spread growth and improve consistency between ensemble spread and forecast error, thereby offering theoretical insights and practical guidance for uncertainty modeling in numerical weather prediction systems.

This work addresses the persistent challenge in process systems modeling of simultaneously achieving accuracy, simplicity, and physical interpretability—particularly in control applications where nonlinear expressiveness must be balanced against a preference for linear structures. The authors propose a convex hybrid modeling paradigm grounded in operator theory, which constrains models to interpretable subspaces or nonlinearly parameterized interpretable manifolds. By introducing a reparameterization technique based on “canonical features” in an augmented parameter space, the approach effectively integrates kernel methods with convex optimization. This framework enables the construction of kernel-based hybrid surrogate models over families of interpretable static and dynamic systems, significantly enhancing both predictive accuracy and computational efficiency while preserving physical interpretability across diverse process systems modeling scenarios.

This work addresses the limitations of traditional generalization analyses, which rely on the often unverifiable assumption of independent and identically distributed (i.i.d.) data and thus struggle to accurately characterize model performance on unseen data. The paper proposes a deterministic generalization analysis framework that dispenses with any prior probabilistic assumptions. By examining the sensitivity of optimization solutions to data perturbations, it decomposes the generalization error into geometric and probabilistic components, achieving their first-ever decoupling. The framework expresses generalization bounds via a variational principle, leveraging deterministic perturbation analysis and optimization sensitivity theory to capture the discrepancy between in-sample and out-of-sample performance. Error terms are evaluated through posterior statistical hypotheses, enabling the recovery of conventional high-probability or expected generalization guarantees—all without requiring distributional assumptions.

Existing Bayesian optimization methods lack theoretical guarantees for adaptive data acquisition in nonlinearly parameterized models. This work proposes an analytical framework based on the reproducing kernel Hilbert space (RKHS) induced by kernels over the parameter space, integrated with regularized convex loss minimization, to establish a unified confidence bound theory for widely used nonlinear surrogate models. For the first time, this framework provides rigorous convergence guarantees for nonlinearly parameterized models under adaptive sampling, enabling a variety of novel acquisition strategies—including stochastic regularization and randomized model maximization—and substantially broadening the theoretical applicability of Bayesian optimization.

This study clarifies the conceptual confusion in physics-oriented machine learning between “interpretability”—referring to model transparency—and “explainability,” which denotes the capacity to map onto domain knowledge. It delineates the boundaries of these two notions and examines their trade-offs in terms of expressive power and adaptability. Through conceptual analysis and the construction of a unifying framework, complemented by a systematic review of both intrinsic and post-hoc explanation methods, the work advocates for integrating interpretability and explainability into scientific modeling paradigms. Crucially, it underscores the central role of task formulation and intervention design in model development. By establishing a clear conceptual foundation and methodological guidance, this research advances the principled integration of machine learning models with scientific reasoning in physics.