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

TL;DR

This paper introduces a hybrid machine learning and homotopy analysis method framework to efficiently compute convergent series solutions for Stokes waves of arbitrary height in deep water, overcoming traditional difficulties near limiting wave conditions.

Contribution

It presents a novel hybrid approach combining HAM with neural networks to rapidly predict series solutions across all wave heights, including limiting waves, with high accuracy and efficiency.

Findings

Neural network trained on 20 cases predicts series solutions across wave heights.

Framework accurately captures solutions up to the theoretical wave limit.

Significantly reduces computational effort compared to traditional methods.

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 $(\theta, r)$ 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.