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

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 (7)

• 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.
• Figure 3: Parametric rolling already appears---and is predicted---in our earlier functional-learning study RoyProccA_JAF, 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.
• 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 $f_e \approx 1/T_e \approx 0.167\,\mathrm{Hz} \approx 2 f_\phi$; 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.