Surrogates for Physics-based and Data-driven Modelling of Parametric Systems: Review and New Perspectives

Abstract: Surrogate models provide compact relations between user-defined input parameters and output quantities of interest, enabling the efficient evaluation of complex parametric systems in many-query settings. Such capabilities are essential in a wide range of applications, including optimisation, control, data assimilation, uncertainty quantification, and emerging digital twin technologies in various fields such as manufacturing, personalised healthcare, smart cities, and sustainability. This article reviews established methodologies for constructing surrogate models exploiting either knowledge of the governing laws and the dynamical structure of the system (physics-based) or experimental observations (data-driven), as well as hybrid approaches combining these two paradigms. By revisiting the design of a surrogate model as a functional approximation problem, existing methodologies are reviewed in terms of the choice of (i) a reduced basis and (ii) a suitable approximation criterion. The paper reviews methodologies pertaining to the field of Scientific Machine Learning, and it aims at synthesising established knowledge, recent advances, and new perspectives on: dimensionality reduction, physics-based, and data-driven surrogate modelling based on proper orthogonal decomposition, proper generalised decomposition, and artificial neural networks; multi-fidelity methods to exploit information from sources with different fidelities; adaptive sampling, enrichment, and data augmentation techniques to enhance the quality of surrogate models.

• The paper introduces a unified framework that casts surrogate construction as a high-dimensional functional approximation problem.
• It compares physics-based, data-driven, and hybrid methods, detailing trade-offs in computational cost, scalability, and interpretability.
• It emphasizes the importance of multi-fidelity strategies and adaptive sampling while outlining open challenges for future surrogate modeling research.

Surrogates for Physics-based and Data-driven Modelling of Parametric Systems: An Authoritative Overview

Introduction

The reviewed work, "Surrogates for Physics-based and Data-driven Modelling of Parametric Systems: Review and New Perspectives" (2603.12870), presents a systematic and comprehensive investigation into methodologies for constructing surrogate models in parametric systems. The paper addresses both the theoretical and practical aspects of surrogate model development, integrating perspectives from physics-based, data-driven, and hybrid approaches. It emphasizes unification via the lens of functional approximation, facilitating cross-disciplinary synthesis and comparison.

Functional Approximation as a Unifying Framework

The foundational premise is the generalization of surrogate model construction as a high-dimensional functional approximation problem. This abstraction clarifies distinctions between the selection of reduced bases (e.g., via POD, PGD, or neural architectures) and the choice of approximation criterion (e.g., interpolation, regression, probabilistic surrogates). By formally treating surrogate construction as the task of approximating a map from parameter space $\mathcal{P}$ to observable quantities, the authors analytically separate the influence of prior information, available data, approximation objectives, and basis representation.

Methods for Surrogate Construction

Reduced Basis Techniques

• Proper Orthogonal Decomposition (POD): Leveraging singular value decomposition for economizing snapshot data, POD forms the basis of both intrusive (POD-RB) and non-intrusive (PODI) models. POD-RB depends on projection of the governing equations onto the reduced basis space, yielding strong interpretability but requiring solver access. PODI focuses on interpolating or regressing the reduced coefficients, enabling black-box application.
• Proper Generalized Decomposition (PGD): PGD seeks explicit parametric separability via greedy identification of rank-one tensor terms. Both a posteriori (from data; non-intrusive) and a priori (operator-level; generally intrusive) formulations are discussed, including their relative computational implications and limitations with respect to system nonlinearity and separability.

Data-driven and Machine Learning Approaches

• Artificial Neural Networks (ANNs): The universal approximation property of feed-forward NNs is exploited for direct functional mapping. Recent advances include architectures explicitly encoding physics (via inductive bias) or learning physics-informed mappings (via learning bias), such as in DeepONet and FNO frameworks.
• Autoencoders and Nonlinear Dimensionality Reduction: Autoencoders provide nonlinear manifold-based reductions, addressing principal deficiencies in the expressiveness of linear methods. The paper presents strategies for both direct surrogate prediction in latent space and hybrid approaches intertwining autoencoders with parametric map learning.
• Graph Neural Networks (GNNs): Proposed as natural surrogates for problems with unstructured spatial discretization, GNNs facilitate the integration of geometry and mesh structure, outperforming classical approaches for unstructured or adaptive mesh data.

Multi-fidelity and Data Assimilation Strategies

• Multi-fidelity Surrogate Models: The synthesis and fusion of high-fidelity (HiFi) and low-fidelity (LoFi) data are systematically covered. Kriging/co-Kriging serve as statistical interpolation baselines, while various multi-level and multi-index approaches extend these ideas using hierarchies of mesh, model, or resolution.
• Multi-fidelity in Regression and Deep Learning: Bayesian and deterministic multi-fidelity regression frameworks, including NARGP, support vector regression, and deep multi-branch or transfer learning NNs are examined in depth.
• Adaptive Sampling and Data Augmentation: The selection of informative samples, via both statistical and adaptive (e.g., DEIM, greedy error estimators) algorithms, is given a full treatment. Data augmentation for enriching training sets—while maintaining physical plausibility—is explicitly detailed.

Methodological Tradeoffs and Performance Criteria

The authors critique each class of surrogate model according to intrusiveness, data requirements, computational cost (offline/online), scalability, expressiveness, robustness to parametric variability, and interpretability. Physics-based approaches (e.g., POD-RB, PGD-a priori) offer strong theoretical underpinnings and stability but can be limited in expressiveness and industrial applicability due to solver access requirements. In contrast, data-driven and deep learning-based models present enhanced flexibility and scalability but often suffer from deficient interpretability, reliability when extrapolating, and dependence on large, high-quality datasets.

Multi-fidelity methods strengthen data efficiency and surrogate reliability by leveraging multiple sources of information; their effectiveness, however, hinges on the strength and structure of inter-model correlations.

Open Problems and Future Perspectives

The authors delineate several outstanding challenges that will drive methodological research and practical adoption of surrogate models:

• Scalability: Surrogate construction for high-dimensional parameter spaces and large physical domains still faces computational and memory bottlenecks.
• Generalization and Robustness: Models must be able to accurately predict outside the training regime, manage uncertainty calibration, and integrate heterogeneous, possibly non-hierarchical, data.
• Interpretability and Explainability: As surrogates become black-boxes, the need for interpretable latent spaces and attribution-based explainability becomes critical, especially in decision-critical industrial and digital twin applications.
• Emerging Paragidms: The intersection with continual learning, online adaptation, transfer/meta-learning, and edge/cloud computing frameworks provides a roadmap for the integration of surrogates into real-time, feedback-driven digital twins.

Conclusion

This work critically surveys and synthesizes a wide swath of the surrogate modelling literature, building a unified, methodological theory grounded in functional approximation. It clarifies methodological tradeoffs and points toward strategic integration of classical and modern (machine learning-based) surrogate approaches. The implications extend across computational science, engineering, and digital twin practice, offering both a blueprint for current best practices and a primer on the critical open questions that will shape the near-future landscape of surrogate-based scientific computing (2603.12870).