# Stochastic stiffness identification and response estimation of Timoshenko beams via physics-informed Gaussian processes

arXiv: https://arxiv.org/abs/2309.11875 · HTML: https://mlsanity.com/paper/2309.11875/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 59 references that are in the map.
- First posted on arXiv 21 Sep 2023.

## Builds on (most cited first, 12 of 12)

- [Gaussian Processes for Big Data](https://mlsanity.com/paper/1309.6835/) (2013)
- [Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1708.00588/) (2017)
- [Machine Learning of Linear Differential Equations using Gaussian Processes](https://mlsanity.com/paper/1701.02440/) (2017)
- [Structure Discovery in Nonparametric Regression through Compositional Kernel Search](https://mlsanity.com/paper/1302.4922/) (2013)
- [Physics informed neural networks for continuum micromechanics](https://mlsanity.com/paper/2110.07374/) (2021)
- [Approximate Inference for Fully Bayesian Gaussian Process Regression](https://mlsanity.com/paper/1912.13440/) (2019)
- [Meshless physics-informed deep learning method for three-dimensional solid mechanics](https://mlsanity.com/paper/2012.01547/) (2020)
- [A Gaussian process latent force model for joint input-state estimation in linear structural systems](https://mlsanity.com/paper/1904.00093/) (2019)
- [Gaussian Process Regression constrained by Boundary Value Problems](https://mlsanity.com/paper/2012.11857/) (2020)
- [A Latent Restoring Force Approach to Nonlinear System Identification](https://mlsanity.com/paper/2109.10681/) (2021)
- [Physics-informed Gaussian process model for Euler-Bernoulli beam elements](https://mlsanity.com/paper/2308.02894/) (2023)
- [Data-driven Aerodynamic Analysis of Structures using Gaussian Processes](https://mlsanity.com/paper/2103.13877/) (2021)

## Built on by (most cited first, 5 of 5)

- [Efficient dynamic modal load reconstruction using physics-informed Gaussian processes based on frequency-sparse Fourier basis functions](https://mlsanity.com/paper/2503.09418/) (2025)
- [Stochastic Inference of Plate Bending from Heterogeneous Data: Physics-informed Gaussian Processes via Kirchhoff-Love Theory](https://mlsanity.com/paper/2405.12802/) (2024)
- [Navigating Uncertainties in Machine Learning for Structural Dynamics: A Comprehensive Survey of Probabilistic and Non-Probabilistic Approaches in Forward and Inverse Problems](https://mlsanity.com/paper/2408.08629/) (2024)
- [Aerodynamic force reconstruction using physics-informed Gaussian processes](https://mlsanity.com/paper/2605.22111/) (2026)
- [Modelling non-linear aeroelastic loads in long-span bridges with extreme learning machines](https://mlsanity.com/paper/2609.33274/) (2026)

## Abstract (arXiv)

Machine learning models trained with structural health monitoring data have become a powerful tool for system identification. This paper presents a physics-informed Gaussian process (GP) model for Timoshenko beam elements. The model is constructed as a multi-output GP with covariance and cross-covariance kernels analytically derived based on the differential equations for deflections, rotations, strains, bending moments, shear forces and applied loads. Stiffness identification is performed in a Bayesian format by maximising a posterior model through a Markov chain Monte Carlo method, yielding a stochastic model for the structural parameters. The optimised GP model is further employed for probabilistic predictions of unobserved responses. Additionally, an entropy-based method for physics-informed sensor placement optimisation is presented, exploiting heterogeneous sensor position information and structural boundary conditions built into the GP model. Results demonstrate that the proposed approach is effective at identifying structural parameters and is capable of fusing data from heterogeneous and multi-fidelity sensors. Probabilistic predictions of structural responses and internal forces are in closer agreement with measured data. We validate our model with an experimental setup and discuss the quality and uncertainty of the obtained results. The proposed approach has potential applications in the field of structural health monitoring (SHM) for both mechanical and structural systems.