Convergent series of Stokes wave of arbitrary height in deep water via machine learning

TL;DR

The paper addresses the challenge of obtaining convergent Stokes-wave series for arbitrary wave heights in deep water, where crest singularities hinder traditional perturbation methods. It develops a HAM-ML hybrid framework: HAM computes accurate series at a sparse set of heights while a neural network generalizes to all steepness values, and a second network learns the inverse conformal mapping to express results in the physical plane. The approach yields high-fidelity predictions across the full $0 \,\leq\ A/L \,\le\,0.14108$ range, including the limiting wave with a $120^ ext{o}$ crest, and delivers substantial speedups (millisecond predictions) without sacrificing symbolic consistency. This work offers a generalizable paradigm for fast, convergent-series solutions in strongly nonlinear systems and paves the way for integrating PINNs to further enhance physical fidelity and applicability.

Abstract

Permanent gravity waves propagating in deep water, spanning amplitudes from infinitesimal to their theoretical limiting values, remain a classical yet challenging problem due to its inherent nonlinear complexities. Traditional analytical and numerical methods encounter substantial difficulties near the limiting wave condition due to singularities at sharp wave crests. In this study, we propose a novel hybrid framework combining the homotopy analysis method (HAM) with machine learning (ML) to efficiently compute convergent series solutions of Stokes waves in deep water for arbitrary wave amplitudes from small to theoretical limiting values, which show excellent agreement with established benchmarks. We introduce a neural network trained using only 20 representative cases whose series solution are given by means of HAM, which can rapidly predict series solutions across arbitrary steepness levels, substantially improving computational efficiency. Additionally, we develop a neural network to gain the inverse mapping from the conformal coordinates $(θ, r)$ to the physical coordinates $(x,y)$, facilitating explicit and intuitive representations of series solutions in physical plane. This HAM-ML hybrid framework represents a powerful and efficient approach to compute convergent series in a whole range of physical parameters for water waves with arbitrary wave height including even limiting waves. In this way we establish a new paradigm to quickly obtain convergent series solutions of complex nonlinear systems for a whole range of physical parameters, thereby significantly broadening the scope of series solutions that can be easily gained by means of HAM even for highly nonlinear problems in science and engineering.

Figures (11)

• Figure 1: (a) $z$ plane, (b) $\zeta$ plane.
• Figure 2: Convergence behavior of selected Fourier coefficients $a_1$, $a_5$, $a_{10}$, $a_{15}$, $a_{20}$, $a_{25}$, $a_{30}$, and the Bernoulli constant $K$ for a Stokes wave in infinite depth with steepness $A/L = 0.07$, based on 1000th-order HAM approximations. The y-axis is the successive order difference $|a_{j,n+1}-a_{j,n}|$ for coefficient $a_j$, and the x-axis is the HAM order $n$.
• Figure 3: Stokes wave profiles in infinite depth for different wave steepness values.
• Figure 4: Comparison of limiting Stokes wave profiles in infinite depth water obtained by different methods.
• Figure 5: Stokes wave flow fields of varying steepness in infinite-depth water. Arrows represent the local velocity vectors, while the color contours depict the magnitude of the flow speed.