Predicting Ship Responses in Different Seaways using a Generalizable Force Correcting Machine Learning Method

TL;DR

This work addresses the challenge of predicting ship responses in diverse seaways with limited high-fidelity data by introducing a neural-corrector hybrid method that augments a low-fidelity equation of motion with a data-driven force correction. The method is evaluated on two case studies: a nonlinear Duffing oscillator with irregular excitation and high-fidelityPANShip data for a Fast Displacement Ship in head seas, and is benchmarked against a linear physics-based model and a data-driven LSTM baseline. Key findings show that embedding physics in the low-fidelity term improves generalizability across unseen sea conditions and that small training datasets (roughly 50–100 zero-up-crossings) can suffice when using simple neural corrections. The approach yields robust distributional predictions, particularly in the tails, and offers a practical surrogate for design evaluations where high-fidelity data are expensive to obtain.

Abstract

A machine learning (ML) method is generalizable if it can make predictions on inputs which differ from the training dataset. For predictions of wave-induced ship responses, generalizability is an important consideration if ML methods are to be useful in design evaluations. Furthermore, the size of the training dataset has a significant impact on the practicality of a method, especially when training data is generated using high-fidelity numerical tools which are expensive. This paper considers a hybrid machine learning method which corrects the force in a low-fidelity equation of motion. The method is applied to two different case studies: the nonlinear responses of a Duffing equation subject to irregular excitation, and high-fidelity heave and pitch response data of a Fast Displacement Ship (FDS) in head seas. The generalizability of the method is determined in both cases by making predictions of the response in irregular wave conditions that differ from those in the training dataset. The influence that low-fidelity physics-based terms in the hybrid model have on generalizability is also investigated. The predictions are compared to two benchmarks: a linear physics-based model and a data-driven LSTM model. It is found that the hybrid method offers an improvement in prediction accuracy and generalizability when trained on a small dataset.

Figures (17)

• Figure 1: An example of high-fidelity responses $z^{(h)}$, $\dot{z}^{(h)}$, $\ddot{z}^{(h)}$ computed using the Duffing equation for use as training data. The wave elevation $\eta$ comes from Eq. \ref{['eq:Sw']} where $H_{s}$ = 1.0 m, $\omega_{p}$ = 1.0 rad/s, and the total number of Zero-Up-Crossings (ZUC) is 108.
• Figure 2: An example of the force correction $\delta$ computed using different low-fidelity forcing models $f^{(l)}$ for use as training data, waves are $H_{s}$ = 1.0 m, $\omega_{p}$ = 1.0 rad/s. For clarity, only time series from 200 s to 300 s is shown, but the pdfs encompass the entire 500 s time series.
• Figure 3: $L_{2}$, $L_{\infty}$, and $\mathrm{JSD}$ prediction errors vs training data set size for each of the five low-fidelity forcing models $f^{(l)}$ in Table \ref{['tab:dufffl']}. The training data set is measured in terms of wave Zero-Up-Crossings (ZUCs). The testing waves come from Eq. \ref{['eq:Sw']} where $H_{s}$ = 1.0 m, $\omega_{p}$ = 1.0 rad/s, and responses are solved using $\Delta t$ = 0.1 s with a stencil length $k$ = 5.
• Figure 4: $L_{2}$, $L_{\infty}$, and $\mathrm{JSD}$ prediction errors vs $H_{s}$ for each of the five low-fidelity forcing models $f^{(l)}$ in Table \ref{['tab:dufffl']}. The training wave condition is shown with a vertical dashed line corresponding to $H_{s}$ = 1 m.
• Figure 5: Predicted responses for proposed model $*$ and LSTM model compared to the target high-fidelity data $h$ in the smallest $H_{s}$ = 0.01 m and largest $H_{s}$ = 1.5 m waves in the testing data set. Training waves: $H_{s}$ = 1.0 m. Only 200 s of time series are shown for clarity, however, the pdfs include the entire 900 s of data.