# A Gaussian process latent force model for joint input-state estimation in linear structural systems

arXiv: https://arxiv.org/abs/1904.00093 · HTML: https://mlsanity.com/paper/1904.00093/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Mar 2019.

## Built on by (most cited first, 18 of 18)

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- [On the Integration of Physics-Based Machine Learning with Hierarchical Bayesian Modeling Techniques](https://mlsanity.com/paper/2303.00187/) (2023)
- [Efficient dynamic modal load reconstruction using physics-informed Gaussian processes based on frequency-sparse Fourier basis functions](https://mlsanity.com/paper/2503.09418/) (2025)
- [Digital twin for virtual sensing of ferry quays via a Gaussian Process Latent Force Model](https://mlsanity.com/paper/2506.14925/) (2025)
- [Deep recurrent-convolutional neural network learning and physics Kalman filtering comparison in dynamic load identification](https://mlsanity.com/paper/2511.00100/) (2025)
- [Machine learning assisted state prediction of misspecified linear dynamical system via modal reduction](https://mlsanity.com/paper/2601.05297/) (2026)
- [CoNBONet: Conformalized Neuroscience-inspired Bayesian Operator Network for Reliability Analysis](https://mlsanity.com/paper/2603.21678/) (2026)

## Abstract (arXiv)

The problem of combined state and input estimation of linear structural systems based on measured responses and a priori knowledge of structural model is considered. A novel methodology using Gaussian process latent force models is proposed to tackle the problem in a stochastic setting. Gaussian process latent force models (GPLFMs) are hybrid models that combine differential equations representing a physical system with data-driven non-parametric Gaussian process models. In this work, the unknown input forces acting on a structure are modelled as Gaussian processes with some chosen covariance functions which are combined with the mechanistic differential equation representing the structure to construct a GPLFM. The GPLFM is then conveniently formulated as an augmented stochastic state-space model with additional states representing the latent force components, and the joint input and state inference of the resulting model is implemented using Kalman filter. The augmented state-space model of GPLFM is shown as a generalization of the class of input-augmented state-space models, is proven observable, and is robust compared to conventional augmented formulations in terms of numerical stability. The hyperparameters governing the covariance functions are estimated using maximum likelihood optimization based on the observed data, thus overcoming the need for manual tuning of the hyperparameters by trial-and-error. To assess the performance of the proposed GPLFM method, several cases of state and input estimation are demonstrated using numerical simulations on a 10-dof shear building and a 76-storey ASCE benchmark office tower. Results obtained indicate the superior performance of the proposed approach over conventional Kalman filter based approaches.