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Discussing the Spectrum of Physics-Enhanced Machine Learning; a Survey on Structural Mechanics Applications

Marcus Haywood-Alexander, Wei Liu, Kiran Bacsa, Zhilu Lai, Eleni Chatzi

TL;DR

The spectrum of PEML methods, expressed across the defining axes of physics and data, is discussed by engaging in a comprehensive exploration of its characteristics, usage, and motivations.

Abstract

The intersection of physics and machine learning has given rise to the physics-enhanced machine learning (PEML) paradigm, aiming to improve the capabilities and reduce the individual shortcomings of data- or physics-only methods. In this paper, the spectrum of physics-enhanced machine learning methods, expressed across the defining axes of physics and data, is discussed by engaging in a comprehensive exploration of its characteristics, usage, and motivations. In doing so, we present a survey of recent applications and developments of PEML techniques, revealing the potency of PEML in addressing complex challenges. We further demonstrate application of select such schemes on the simple working example of a single degree-of-freedom Duffing oscillator, which allows to highlight the individual characteristics and motivations of different `genres' of PEML approaches. To promote collaboration and transparency, and to provide practical examples for the reader, the code generating these working examples is provided alongside this paper. As a foundational contribution, this paper underscores the significance of PEML in pushing the boundaries of scientific and engineering research, underpinned by the synergy of physical insights and machine learning capabilities.

Discussing the Spectrum of Physics-Enhanced Machine Learning; a Survey on Structural Mechanics Applications

TL;DR

The spectrum of PEML methods, expressed across the defining axes of physics and data, is discussed by engaging in a comprehensive exploration of its characteristics, usage, and motivations.

Abstract

The intersection of physics and machine learning has given rise to the physics-enhanced machine learning (PEML) paradigm, aiming to improve the capabilities and reduce the individual shortcomings of data- or physics-only methods. In this paper, the spectrum of physics-enhanced machine learning methods, expressed across the defining axes of physics and data, is discussed by engaging in a comprehensive exploration of its characteristics, usage, and motivations. In doing so, we present a survey of recent applications and developments of PEML techniques, revealing the potency of PEML in addressing complex challenges. We further demonstrate application of select such schemes on the simple working example of a single degree-of-freedom Duffing oscillator, which allows to highlight the individual characteristics and motivations of different `genres' of PEML approaches. To promote collaboration and transparency, and to provide practical examples for the reader, the code generating these working examples is provided alongside this paper. As a foundational contribution, this paper underscores the significance of PEML in pushing the boundaries of scientific and engineering research, underpinned by the synergy of physical insights and machine learning capabilities.
Paper Structure (28 sections, 24 equations, 13 figures, 2 tables)

This paper contains 28 sections, 24 equations, 13 figures, 2 tables.

Table of Contents

  1. Introduction
  2. A Working Example
  3. A note on data and domains
  4. White Box Case - Physics-based Bayesian Filtering
  5. The Black Box Case - Deep Learning Models
  6. Light Grey PEML Schemes
  7. ML-Enhanced Bayesian Filtering
  8. Physics-Guided Neural Networks
  9. Grey PEML Schemes
  10. Dictionary Methods
  11. Physics-Informed Neural Networks
  12. PINNs for System Identification
  13. PINNs for Enhanced Learning
  14. PINNs for Forward Modelling
  15. Dark Grey PEML schemes
  16. ...and 13 more sections

Figures (13)

  • Figure 1: The spectrum of Physics-Enhanced Machine Learning (PEML) schemes surveyed in this paper
  • Figure 2: (a) Diagram of the working example used throughout this paper, corresponding to a Duffing Oscillator; instances of the (b) displacement (top) and forcing signal (bottom) produced during simulation
  • Figure 3: Visualisation of domain definitions for schemes and motivations that can employ PEML. The blue areas represent the continuous collocation domain, and the red dots represent the coverage and sparsity of the discrete observation domain. The dashed and solid lines represent the scope of the collocation and observation domains, respectively
  • Figure 4: (a) State (response) estimation results for the nonlinear SDOF working example, assuming availability of acceleration measurements and precise knowledge of the model form, albeit under the assumption of unknown model parameters. The performance is illustrated for use of the UKF and PF, contrasted against the reference simulation; (b) Parameter estimation convergence via use of the UKF and PF contrasted against the reference values for the nonlinear SDOF working example
  • Figure 5: Predicted latent representations vs exact solutions of displacement (top) and velocity (bottom) using the DMM applied to the working example. Displacement is assume to be the only measurement. The blue bounding boxes represent the estimated $2\sigma$ range
  • ...and 8 more figures