Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Matteo Giacomini, Pedro Díez

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.

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.
Paper Structure (37 sections, 53 equations, 2 figures, 2 algorithms)

This paper contains 37 sections, 53 equations, 2 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Parametric full-order models
  3. Surrogates as functional approximations
  4. Verification, validation, and uncertainty quantification
  5. Structure of the manuscript
  6. Revisiting functional approximation criteria
  7. Polynomial interpolation
  8. Non-polynomial interpolation
  9. Least-squares approximation
  10. Moving least-squares approximation
  11. Universal approximation via artificial neural networks
  12. Proper orthogonal decomposition
  13. Dimensionality reduction using POD
  14. PODI
  15. POD-RB
  16. ...and 22 more sections

Figures (2)

  • Figure 1: Illustration of a neural network with one input neuron, three hidden layers with two neurons and one output neuron. In each layer, the blank dot is associated with bias.
  • Figure 2: Illustration of an autoencoder neural network.

Theorems & Definitions (10)

  • Remark 1: Spectral approximations
  • Remark 2: Variogram
  • Remark 3: Choice of the scalar product
  • Remark 4: Optimal sampling points
  • Remark 5: High-order SVD
  • Remark 6: Interpretation of the parametric modes
  • Remark 7: Data availability
  • Remark 8: Multi-fidelity surrogates with interpolation
  • Remark 9: Multi-fidelity surrogates with regression
  • Remark 10: Multi-fidelity surrogates with NN