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

arXiv: https://arxiv.org/abs/2310.20425 · HTML: https://mlsanity.com/paper/2310.20425/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 55 of its 166 references that are in the map.
- First posted on arXiv 31 Oct 2023.

## Builds on (most cited first, 30 of 55)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [Symplectic ODE-Net: Learning Hamiltonian Dynamics with Control](https://mlsanity.com/paper/1909.12077/) (2019)
- [Structured Inference Networks for Nonlinear State Space Models](https://mlsanity.com/paper/1609.09869/) (2016)
- [Symplectic Recurrent Neural Networks](https://mlsanity.com/paper/1909.13334/) (2019)
- [Sparse identification of nonlinear dynamics for model predictive control in the low-data limit](https://mlsanity.com/paper/1711.05501/) (2017)
- [KalmanNet: Neural Network Aided Kalman Filtering for Partially Known Dynamics](https://mlsanity.com/paper/2107.10043/) (2021)
- [Physics-guided Neural Networks (PGNN): An Application in Lake Temperature Modeling](https://mlsanity.com/paper/1710.11431/) (2017)
- [Transfer learning enhanced physics informed neural network for phase-field modeling of fracture](https://mlsanity.com/paper/1907.02531/) (2019)
- [Hamiltonian Graph Networks with ODE Integrators](https://mlsanity.com/paper/1909.12790/) (2019)
- [Hilbert Space Methods for Reduced-Rank Gaussian Process Regression](https://mlsanity.com/paper/1401.5508/) (2014)
- [Multiscale Simulations of Complex Systems by Learning their Effective Dynamics](https://mlsanity.com/paper/2006.13431/) (2020)
- [The relationship between trust in AI and trustworthy machine learning technologies](https://mlsanity.com/paper/1912.00782/) (2019)
- [Can Physics-Informed Neural Networks beat the Finite Element Method?](https://mlsanity.com/paper/2302.04107/) (2023)
- [Port-Hamiltonian Neural Networks for Learning Explicit Time-Dependent Dynamical Systems](https://mlsanity.com/paper/2107.08024/) (2021)
- [Learning Stochastic Recurrent Networks](https://mlsanity.com/paper/1411.7610/) (2014)
- [Physics Guided RNNs for Modeling Dynamical Systems: A Case Study in Simulating Lake Temperature Profiles](https://mlsanity.com/paper/1810.13075/) (2018)
- [Unsupervised discovery of interpretable hyperelastic constitutive laws](https://mlsanity.com/paper/2010.13496/) (2020)
- [Dissipative Hamiltonian Neural Networks: Learning Dissipative and Conservative Dynamics Separately](https://mlsanity.com/paper/2201.10085/) (2022)
- [Data-Driven Deep Learning of Partial Differential Equations in Modal Space](https://mlsanity.com/paper/1910.06948/) (2019)
- [Physics informed neural networks for continuum micromechanics](https://mlsanity.com/paper/2110.07374/) (2021)
- [Physics-Informed Neural Networks for Nonhomogeneous Material Identification in Elasticity Imaging](https://mlsanity.com/paper/2009.04525/) (2020)
- [Variational Integrator Networks for Physically Structured Embeddings](https://mlsanity.com/paper/1910.09349/) (2019)
- [Discovery of Nonlinear Dynamical Systems using a Runge-Kutta Inspired Dictionary-based Sparse Regression Approach](https://mlsanity.com/paper/2105.04869/) (2021)
- [Symplectic Learning for Hamiltonian Neural Networks](https://mlsanity.com/paper/2106.11753/) (2021)

## Built on by (most cited first, 7 of 7)

- [Response Estimation and System Identification of Dynamical Systems via Physics-Informed Neural Networks](https://mlsanity.com/paper/2410.01340/) (2024)
- [Structured Kolmogorov-Arnold Neural ODEs for Interpretable Learning and Symbolic Discovery of Nonlinear Dynamics](https://mlsanity.com/paper/2506.18339/) (2025)
- [Adversarial Disentanglement by Backpropagation with Physics-Informed Variational Autoencoder](https://mlsanity.com/paper/2506.13658/) (2025)
- [PiGGO: Physics-Guided Learnable Graph Kalman Filters for Virtual Sensing of Nonlinear Dynamic Structures under Uncertainty](https://mlsanity.com/paper/2604.26593/) (2026)
- [Digital twin for virtual sensing of ferry quays via a Gaussian Process Latent Force Model](https://mlsanity.com/paper/2506.14925/) (2025)
- [From Grey-Box to Green-Box: When can Physics-Informed Machine Learning Reduce Carbon Footprints in Structural Health Monitoring?](https://mlsanity.com/paper/2609.33387/) (2026)
- [Dissipative Lagrangian Neural Networks for the Interpretable Modelling of Passive Mechanical Systems](https://mlsanity.com/paper/2610.15996/) (2026)

## Abstract (arXiv)

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.