# On the Integration of Physics-Based Machine Learning with Hierarchical Bayesian Modeling Techniques

arXiv: https://arxiv.org/abs/2303.00187 · HTML: https://mlsanity.com/paper/2303.00187/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 48 references that are in the map.
- First posted on arXiv 1 Mar 2023.

## Builds on (most cited first, 6 of 6)

- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Numerical Gaussian Processes for Time-dependent and Non-linear Partial Differential Equations](https://mlsanity.com/paper/1703.10230/) (2017)
- [A Gaussian process latent force model for joint input-state estimation in linear structural systems](https://mlsanity.com/paper/1904.00093/) (2019)
- [Grey-box models for wave loading prediction](https://mlsanity.com/paper/2105.13813/) (2021)
- [Constraining Gaussian processes for physics-informed acoustic emission mapping](https://mlsanity.com/paper/2206.01495/) (2022)
- [Hierarchical Bayesian Operational Modal Analysis: Theory and Computations](https://mlsanity.com/paper/1908.06370/) (2019)

## Built on by (most cited first, 3 of 3)

- [On the hierarchical Bayesian modelling of frequency response functions](https://mlsanity.com/paper/2307.06263/) (2023)
- [Physics-informed neural networks for parameter learning of wildfire spreading](https://mlsanity.com/paper/2406.14591/) (2024)
- [Meta-models for transfer learning in source localisation](https://mlsanity.com/paper/2305.08657/) (2023)

## Abstract (arXiv)

Machine Learning (ML) has widely been used for modeling and predicting physical systems. These techniques offer high expressive power and good generalizability for interpolation within observed data sets. However, the disadvantage of black-box models is that they underperform under blind conditions since no physical knowledge is incorporated. Physics-based ML aims to address this problem by retaining the mathematical flexibility of ML techniques while incorporating physics. In accord, this paper proposes to embed mechanics-based models into the mean function of a Gaussian Process (GP) model and characterize potential discrepancies through kernel machines. A specific class of kernel function is promoted, which has a connection with the gradient of the physics-based model with respect to the input and parameters and shares similarity with the exact Autocovariance function of linear dynamical systems. The spectral properties of the kernel function enable considering dominant periodic processes originating from physics misspecification. Nevertheless, the stationarity of the kernel function is a difficult hurdle in the sequential processing of long data sets, resolved through hierarchical Bayesian techniques. This implementation is also advantageous to mitigate computational costs, alleviating the scalability of GPs when dealing with sequential data. Using numerical and experimental examples, potential applications of the proposed method to structural dynamics inverse problems are demonstrated.