Learning Response-Statistic Shifts and Parametric Roll Episodes from Wave--Vessel Time Series via LSTM Functional Models

Jose del Aguila Ferrandis

cs.LGMar 25, 2026

Learning Response-Statistic Shifts and Parametric Roll Episodes from Wave--Vessel Time Series via LSTM Functional Models

Authors: Jose del Aguila Ferrandis

Organizations: Kevin T. Crofton Department of Aerospace and Ocean Engineering, Virginia Tech, USA

Abstract

Parametric roll is a rare but high-consequence instability that can trigger abrupt regime changes in ship response, including pronounced shifts in roll statistics and tail risk. This paper develops a data-driven surrogate that learns the nonlinear, causal functional mapping from incident wave--motion time series to vessel motions, and demonstrates that the surrogate reproduces both (i) parametric roll episodes and (ii) the associated statistical shifts in the response. Crucially, the learning framework is data-source agnostic: the paired wave--motion time series can be obtained from controlled experiments (e.g., towing-tank or basin tests with wave probes and motion tracking) when a hull exists, or from high-fidelity simulations during design when experiments are not yet available. To provide a controlled severe-sea demonstration, we generate training data with a URANS numerical wave tank, using long-crested irregular seas synthesized from a modified Pierson--Moskowitz spectrum. The demonstration dataset comprises 49 random-phase realizations for each of three sea states, simulated at a fixed forward speed selected to yield encounter conditions under which parametric-roll episodes can occur. A stacked LSTM surrogate is trained on wave-elevation time series and evaluated on held-out realizations using time-domain accuracy and distributional fidelity metrics. In the most severe case, the model tracks the onset and growth of large-amplitude roll consistent with parametric excitation, and captures the corresponding changes in roll probability density functions (PDFs). We further compare loss-function choices (MSE, relative-entropy-based objectives, and amplitude-weighted variants) and show how they trade average error for improved tail fidelity relevant to operability and risk assessment.

Figures & tables

Case	Significant height $H_{s}$ [m]	Peak period $T_{p}$ [s]
SS–1	3.53	9.7
SS–2	5.09	12.4
SS–3	10.66	13.4

Table 1: Sea states in the URANS campaign (49 random-phase seeds per sea state). Across the full campaign (three sea states, 49 random-phase realizations each), the total simulated duration of each ensemble exceeds

3600\text{\,}\mathrm{s}

Panel of Figure 1 — Figure 1: Inputs for the spatiotemporal stencil: (left) probe layout; (right) representative wave profiles in longitudinal and transversal directions.

Figure 2: Roll response-statistics fidelity in the most severe sea state (SS–3): comparison of observed URANS roll PDF to learned surrogate-induced PDFs under different training objectives. To show the departure from gaussian statistics we include a fit with a gaussian distribution of the URANS roll PDF.

Panel of Figure 3 — Figure 3: Parametric rolling already appears—and is predicted—in our earlier functional-learning study [ 11 ] , as evidenced by the roll–pitch phase portrait and CFD-versus-LSTM comparisons on held-out realizations. However, because that dataset was limited to a single sea state, it did not test whether the learned functional captures regime-dependent shifts in response statistics. Here we extend that work by pooling realizations across distinct forcing conditions and not conditioning on sea-state labels (e.g., $H_{s}$ , $T_{p}$ ), so any changes in predicted PDFs/tails must be inferred directly from the raw wave-elevation history.

Panel of Figure 4 — Figure 4: Representative held-out time-series comparisons for seed 41 across the three sea states. Each panel shows heave, pitch, and roll responses, comparing the URANS reference (CFD) against the learned surrogates trained with different objectives. Across all sea states, the surrogate predictions capture the dominant oscillatory structure and timing.

Figure 5: Short-time Fourier transform (STFT) magnitude of the H , P , and R signals (frequency in Hz), highlighting a transition interval (red boxes; $\approx 60$ – $80$ s) where energy becomes more concentrated in the response band. In the parametric-roll scenario described in the text, a vessel with $T_{\phi}\approx 12\,\mathrm{s}$ corresponds to $f_{\phi}\approx 1/T_{\phi}\approx 0.083\,\mathrm{Hz}$ , and repeated encounters at $T_{e}\approx 6\,\mathrm{s}$ correspond to $f_{e}\approx 1/T_{e}\approx 0.167\,\mathrm{Hz}\approx 2f_{\phi}$ ; the highlighted interval is consistent with the emergence/strengthening of a narrow-band component near the roll-response frequency and its associated parametric excitation. Color denotes STFT amplitude.

Panel of Figure 6 — Figure 6: Reference roll response statistics across sea states. Each panel shows the predicted roll probability density (PDF) for a held-out ensemble in SS–1/SS–2/SS–3. The distribution remains approximately Gaussian in SS–1 and SS–2, while SS–3 exhibits pronounced non-Gaussian tail behavior consistent with intermittent large-roll episodes (parametric-roll regime).

Panel of Figure 7 — Figure 7: Averaged LSTM test results (6 hidden layers, 250 neurons). Bar charts show the average of the Heave, Pitch, and Roll absolute-difference metrics (MSE, RE, and AWMSE) over the 9 testing seeds for a notional DTMB battleship at Froude number 0.4, time step $\Delta t=0.2\,\mathrm{s}$ , and speed $v=30\,\mathrm{kt}$ , for Sea States 1, 2, and 3.

Neural Surrogate Modeling

Neural surrogate modeling uses learned models to approximate costly simulations, enabling faster prediction, design, control, and inference.

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Efficient Time-Domain Simulation of USV Motions in Short-Crested Irregular Waves Using an IRF-Based Framework

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Unmanned Surface Vehicle

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Traditional time-domain prediction of vessel motions in irregular waves usually relies on superposing responses from many regular-wave components, which is computationally expensive for long-duration simulation and real-time applications. This issue is particularly relevant to unmanned surface vehicles (USVs), for which efficient and realistic motion prediction is needed for seakeeping assessment, simulation-based testing, and control-system development. This study applies an impulse response function (IRF)-based time-domain framework to predict vessel motions in short-crested irregular waves. Froude-Krylov, diffraction, and radiation loads are obtained from frequency-domain analysis and transformed into the time domain. Instantaneous responses are then evaluated directly through convolution-based force reconstruction, reducing the need for repeated regular-wave simulations. Weak nonlinear restoring effects are included by instantaneous wetted-surface pressure integration, and directional wave spectra are used to represent realistic sea states. The framework is validated against model-test measurements of an offshore supply vessel in long-crested beam irregular waves and full-scale measurements of a USV operating in real sea conditions. Predicted significant amplitudes, mean zero-crossing periods, standard deviations, and motion time histories agree well with measurements. The effect of directional-spectrum discretization is also examined. Results show that motion amplitudes are moderately sensitive to directional resolution, whereas motion periods are relatively insensitive. A 30 deg directional interval provides a practical balance between prediction accuracy and computational cost. The proposed framework offers an efficient tool for high-fidelity time-domain prediction of USV motions in realistic directional irregular seas.

Abstract

Neural Surrogate Modeling

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