# Grey-box models for wave loading prediction

arXiv: https://arxiv.org/abs/2105.13813 · HTML: https://mlsanity.com/paper/2105.13813/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- Builds on 2 of its 42 references that are in the map.
- First posted on arXiv 10 May 2021.

## Builds on (most cited first, 2 of 2)

- [Gaussian Processes for Big Data](https://mlsanity.com/paper/1309.6835/) (2013)
- [Modeling and interpolation of the ambient magnetic field by Gaussian processes](https://mlsanity.com/paper/1509.04634/) (2015)

## Built on by (most cited first, 13 of 13)

- [Improving aircraft performance using machine learning: a review](https://mlsanity.com/paper/2210.11481/) (2022)
- [A spectrum of physics-informed Gaussian processes for regression in engineering](https://mlsanity.com/paper/2309.10656/) (2023)
- [Physics-informed machine learning for Structural Health Monitoring](https://mlsanity.com/paper/2206.15303/) (2022)
- [Constraining Gaussian processes for physics-informed acoustic emission mapping](https://mlsanity.com/paper/2206.01495/) (2022)
- [On generative models as the basis for digital twins](https://mlsanity.com/paper/2203.04384/) (2022)
- [Using scientific machine learning for experimental bifurcation analysis of dynamic systems](https://mlsanity.com/paper/2110.11854/) (2021)
- [On the Integration of Physics-Based Machine Learning with Hierarchical Bayesian Modeling Techniques](https://mlsanity.com/paper/2303.00187/) (2023)
- [Towards a population-informed approach to the definition of data-driven models for structural dynamics](https://mlsanity.com/paper/2307.09862/) (2023)
- [Physics-integrated generative modeling using attentive planar normalizing flow based variational autoencoder](https://mlsanity.com/paper/2404.12267/) (2024)
- [Data-driven Modeling in Metrology -- A Short Introduction, Current Developments and Future Perspectives](https://mlsanity.com/paper/2406.16659/) (2024)
- [A Meta-Learning Approach to Population-Based Modelling of Structures](https://mlsanity.com/paper/2302.07980/) (2023)
- [Equation-Free Digital Twins for Nonlinear Structural Dynamics](https://mlsanity.com/paper/2605.00950/) (2026)
- [Equation discovery with Bayesian tree-adjoining grammars](https://mlsanity.com/paper/2609.31368/) (2026)

## Abstract (arXiv)

The quantification of wave loading on offshore structures and components is a crucial element in the assessment of their useful remaining life. In many applications the well-known Morison's equation is employed to estimate the forcing from waves with assumed particle velocities and accelerations. This paper develops a grey-box modelling approach to improve the predictions of the force on structural members. A grey-box model intends to exploit the enhanced predictive capabilities of data-based modelling whilst retaining physical insight into the behaviour of the system; in the context of the work carried out here, this can be considered as physics-informed machine learning. There are a number of possible approaches to establish a grey-box model. This paper demonstrates two means of combining physics (white box) and data-based (black box) components; one where the model is a simple summation of the two components, the second where the white-box prediction is fed into the black box as an additional input. Here Morison's equation is used as the physics-based component in combination with a data-based Gaussian process NARX - a dynamic variant of the more well-known Gaussian process regression. Two key challenges with employing the GP-NARX formulation that are addressed here are the selection of appropriate lag terms and the proper treatment of uncertainty propagation within the dynamic GP. The best performing grey-box model, the residual modelling GP-NARX, was able to achieve a 29.13% and 5.48% relative reduction in NMSE over Morison's Equation and a black-box GP-NARX respectively, alongside significant benefits in extrapolative capabilities of the model, in circumstances of low dataset coverage.