# A spectrum of physics-informed Gaussian processes for regression in engineering

arXiv: https://arxiv.org/abs/2309.10656 · HTML: https://mlsanity.com/paper/2309.10656/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- Builds on 30 of its 86 references that are in the map.
- First posted on arXiv 19 Sep 2023.

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

- [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1708.00588/) (2017)
- [Informed Machine Learning -- A Taxonomy and Survey of Integrating Knowledge into Learning Systems](https://mlsanity.com/paper/1903.12394/) (2019)
- [Theory-guided Data Science: A New Paradigm for Scientific Discovery from Data](https://mlsanity.com/paper/1612.08544/) (2016)
- [Machine Learning of Linear Differential Equations using Gaussian Processes](https://mlsanity.com/paper/1701.02440/) (2017)
- [Gaussian Process Kernels for Pattern Discovery and Extrapolation](https://mlsanity.com/paper/1302.4245/) (2013)
- [Numerical Gaussian Processes for Time-dependent and Non-linear Partial Differential Equations](https://mlsanity.com/paper/1703.10230/) (2017)
- [Nonlinear System Identification: A User-Oriented Roadmap](https://mlsanity.com/paper/1902.00683/) (2019)
- [A Survey of Constrained Gaussian Process Regression: Approaches and Implementation Challenges](https://mlsanity.com/paper/2006.09319/) (2020)
- [Bayesian Probabilistic Numerical Methods](https://mlsanity.com/paper/1702.03673/) (2017)
- [Convolutional Gaussian Processes](https://mlsanity.com/paper/1709.01894/) (2017)
- [Physics-Informed Gaussian Process Regression Generalizes Linear PDE Solvers](https://mlsanity.com/paper/2212.12474/) (2022)
- [Inferring solutions of differential equations using noisy multi-fidelity data](https://mlsanity.com/paper/1607.04805/) (2016)
- [Spectral Mixture Kernels for Multi-Output Gaussian Processes](https://mlsanity.com/paper/1709.01298/) (2017)
- [Modeling and interpolation of the ambient magnetic field by Gaussian processes](https://mlsanity.com/paper/1509.04634/) (2015)
- [AutoIP: A United Framework to Integrate Physics into Gaussian Processes](https://mlsanity.com/paper/2202.12316/) (2022)
- [A Gaussian process latent force model for joint input-state estimation in linear structural systems](https://mlsanity.com/paper/1904.00093/) (2019)
- [Gaussian processes with linear operator inequality constraints](https://mlsanity.com/paper/1901.03134/) (2019)
- [Gaussian Process Manifold Interpolation for Probabilistic Atrial Activation Maps and Uncertain Conduction Velocity](https://mlsanity.com/paper/2004.10586/) (2020)
- [Grey-box models for wave loading prediction](https://mlsanity.com/paper/2105.13813/) (2021)
- [Know Your Boundaries: Constraining Gaussian Processes by Variational Harmonic Features](https://mlsanity.com/paper/1904.05207/) (2019)
- [Black-Box Inference for Non-Linear Latent Force Models](https://mlsanity.com/paper/1906.09199/) (2019)
- [Compositional Modeling of Nonlinear Dynamical Systems with ODE-based Random Features](https://mlsanity.com/paper/2106.05960/) (2021)
- [Constraining Gaussian processes for physics-informed acoustic emission mapping](https://mlsanity.com/paper/2206.01495/) (2022)
- [A Bayesian methodology for localising acoustic emission sources in complex structures](https://mlsanity.com/paper/2012.11058/) (2020)
- [Probabilistic Inference for Structural Health Monitoring: New Modes of Learning from Data](https://mlsanity.com/paper/2103.01676/) (2021)
- [Structured Machine Learning Tools for Modelling Characteristics of Guided Waves](https://mlsanity.com/paper/2101.01506/) (2021)
- [Learning Nonparametric Volterra Kernels with Gaussian Processes](https://mlsanity.com/paper/2106.05582/) (2021)
- [Physically Meaningful Uncertainty Quantification in Probabilistic Wind Turbine Power Curve Models as a Damage Sensitive Feature](https://mlsanity.com/paper/2209.15579/) (2022)
- [Physics-enhanced Gaussian Process Variational Autoencoder](https://mlsanity.com/paper/2305.09006/) (2023)

## Built on by (most cited first, 14 of 14)

- [Physics-Enhanced Machine Learning: a position paper for dynamical systems investigations](https://mlsanity.com/paper/2405.05987/) (2024)
- [Co-orchestration of Multiple Instruments to Uncover Structure-Property Relationships in Combinatorial Libraries](https://mlsanity.com/paper/2402.02198/) (2024)
- [Measurements with Noise: Bayesian Optimization for Co-optimizing Noise and Property Discovery in Automated Experiments](https://mlsanity.com/paper/2410.02717/) (2024)
- [Data-Driven Prediction and Control of Hammerstein-Wiener Systems with Implicit Gaussian Processes](https://mlsanity.com/paper/2501.15849/) (2025)
- [Physically-informed change-point kernels for structural dynamics](https://mlsanity.com/paper/2506.11625/) (2025)
- [Digital twin for virtual sensing of ferry quays via a Gaussian Process Latent Force Model](https://mlsanity.com/paper/2506.14925/) (2025)
- [Rethinking Physics-Informed Regression Beyond Training Loops and Bespoke Architectures](https://mlsanity.com/paper/2512.13217/) (2025)
- [Physics-Informed Framework for Impact Identification in Aerospace Composites](https://mlsanity.com/paper/2603.28593/) (2026)
- [Green Physics-Informed Machine Learning Models For Structural Health Monitoring](https://mlsanity.com/paper/2604.27638/) (2026)
- [Orthogonal Discrepancy Kernels for Learning with Partial Physics](https://mlsanity.com/paper/2606.21199/) (2026)
- [Synergizing Physically Constrained MCMC and Chemical-Informed Gaussian Processes for Reaction Network Discovery](https://mlsanity.com/paper/2606.23757/) (2026)
- [Physics-Informed Machine Learning Under Small-Data Constraints: Lessons from Abrasive Waterjet Milling](https://mlsanity.com/paper/2607.07863/) (2026)
- [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)

Despite the growing availability of sensing and data in general, we remain unable to fully characterise many in-service engineering systems and structures from a purely data-driven approach. The vast data and resources available to capture human activity are unmatched in our engineered world, and, even in cases where data could be referred to as ``big,'' they will rarely hold information across operational windows or life spans. This paper pursues the combination of machine learning technology and physics-based reasoning to enhance our ability to make predictive models with limited data. By explicitly linking the physics-based view of stochastic processes with a data-based regression approach, a spectrum of possible Gaussian process models are introduced that enable the incorporation of different levels of expert knowledge of a system. Examples illustrate how these approaches can significantly reduce reliance on data collection whilst also increasing the interpretability of the model, another important consideration in this context.