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Combining physics-based and data-driven models: advancing the frontiers of research with Scientific Machine Learning

Alfio Quarteroni, Paola Gervasio, Francesco Regazzoni

TL;DR

The successful application of SciML to the simulation of the human cardiac function, a field of significant socioeconomic importance that poses numerous challenges on both the mathematical and computational fronts.

Abstract

Scientific Machine Learning (SciML) is a recently emerged research field which combines physics-based and data-driven models for the numerical approximation of differential problems. Physics-based models rely on the physical understanding of the problem, subsequent mathematical formulation, and numerical approximation. Data-driven models instead aim to extract relations between input and output data without arguing any causality principle underlining the available data distribution. In recent years, data-driven models have been rapidly developed and popularized. Such a diffusion has been triggered by a huge availability of data, increasingly cheap computing power, and the development of powerful ML algorithms. SciML leverages the physical awareness of physics-based models and the efficiency of data-driven algorithms. With SciML, we can inject physics and mathematical knowledge into ML algorithms. Yet, we can rely on data-driven algorithms' capability to discover complex and nonlinear patterns from data and improve the descriptive capacity of physics-based models. After recalling the mathematical foundations of digital modelling and ML algorithms and presenting the most popular ML architectures, we discuss the great potential of a broad variety of SciML strategies in solving complex problems governed by PDEs. Finally, we illustrate the successful application of SciML to the simulation of the human cardiac function, a field of significant socioeconomic importance that poses numerous challenges on both the mathematical and computational fronts. Despite the robustness and accuracy of physics-based models, certain aspects, such as unveiling constitutive laws for cardiac cells and myocardial material properties, as well as devising efficient reduced order models to dominate the extraordinary computational complexity, have been successfully tackled by leveraging data-driven models.

Combining physics-based and data-driven models: advancing the frontiers of research with Scientific Machine Learning

TL;DR

The successful application of SciML to the simulation of the human cardiac function, a field of significant socioeconomic importance that poses numerous challenges on both the mathematical and computational fronts.

Abstract

Scientific Machine Learning (SciML) is a recently emerged research field which combines physics-based and data-driven models for the numerical approximation of differential problems. Physics-based models rely on the physical understanding of the problem, subsequent mathematical formulation, and numerical approximation. Data-driven models instead aim to extract relations between input and output data without arguing any causality principle underlining the available data distribution. In recent years, data-driven models have been rapidly developed and popularized. Such a diffusion has been triggered by a huge availability of data, increasingly cheap computing power, and the development of powerful ML algorithms. SciML leverages the physical awareness of physics-based models and the efficiency of data-driven algorithms. With SciML, we can inject physics and mathematical knowledge into ML algorithms. Yet, we can rely on data-driven algorithms' capability to discover complex and nonlinear patterns from data and improve the descriptive capacity of physics-based models. After recalling the mathematical foundations of digital modelling and ML algorithms and presenting the most popular ML architectures, we discuss the great potential of a broad variety of SciML strategies in solving complex problems governed by PDEs. Finally, we illustrate the successful application of SciML to the simulation of the human cardiac function, a field of significant socioeconomic importance that poses numerous challenges on both the mathematical and computational fronts. Despite the robustness and accuracy of physics-based models, certain aspects, such as unveiling constitutive laws for cardiac cells and myocardial material properties, as well as devising efficient reduced order models to dominate the extraordinary computational complexity, have been successfully tackled by leveraging data-driven models.

Paper Structure

This paper contains 39 sections, 8 theorems, 142 equations, 30 figures, 2 tables, 11 algorithms.

Table of Contents

  1. Introduction
  2. Digital models
  3. Mathematical models
  4. Numerical models
  5. Data--driven models
  6. Artificial Intelligence
  7. Machine Learning
  8. Machine Learning Tasks
  9. Machine Learning Experience
  10. Machine Learning Performance measurement
  11. Machine Learning Models
  12. Setting of supervised learning and error analysis
  13. Optimization methods for training
  14. Backpropagation
  15. Penalty--based regularization
  16. ...and 24 more sections

Key Result

Theorem 3.1

Cybenko1989 Let $K\subset\mathbb{R}^n$ be a compact set and $\sigma\in{\mathcal{C}}({\mathbb R})$ a discriminatory activation function on $K$. Then, given any real--valued function $\hat{f}$ continuous on $K$ and tolerance $\varepsilon>0$, there exists a shallow FFNN with weights and biases $\mathbf

Figures (30)

  • Figure 1: The abstract framework.
  • Figure 2: Problems, solutions, and errors in digital models
  • Figure 8: The ideal model $\hat{\mathbf f}$ and the really computed model $\hat{\mathbf{f}}^*_{\mathcal{H},S}$
  • Figure 9: Underfitting (left), optimal fitting (centre), and overfitting (right) for a classification task
  • Figure 10: The empirical risk (training error), the expected risk, and the generalization error versus the capacity of the hypothesis space $\mathcal{H}$
  • ...and 25 more figures

Theorems & Definitions (12)

  • Example 3.1
  • Remark 3.1
  • Example 3.2
  • Definition 3.1
  • Theorem 3.1: Cybenko (1989)
  • Theorem 3.2: Lower and upper complexity bound for shallow NNs
  • Theorem 3.3: Yarotsky (2017)
  • Theorem 3.4: Güring, Kutyniok, Petersen (2020)
  • Theorem 3.5: Güring, Kutyniok, Petersen (2020)
  • Theorem 4.1
  • ...and 2 more