Unified convergent series solutions of Stokes waves of arbitrary height in arbitrary depth with machine-learning acceleration

Stokes waves are a fundamental model for nonlinear wave dynamics and play a crucial role in understanding complex ocean phenomena. However, owing to their inherent nonlinearity, accurately and efficiently solving Stokes waves of arbitrary height in arbitrary depth has long remained a challenging problem. In this work, we propose a unified framework combining the homotopy analysis method (HAM) with machine learning to rapidly compute convergent series of Stokes waves across the entire physical parameter space. We introduce an accelerated HAM strategy, utilizing first-order iteration and precomputation, to solve Stokes waves for arbitrary prescribed depth and wave height conditions. Based on this, we develop a multi-stage, multi-scale neural network that maps wave parameters directly to the convergent series coefficients. This architecture effectively manages strong coefficient scale separation and preserves the high-order modes critical for resolving limiting crests, computing a Stokes wave in ∼0.01 s. Additionally, a Fourier neural operator (FNO) is employed to approximate the inverse conformal mapping, enabling the rapid reconstruction of physical quantities, such as velocity fields, in the physical plane. Therefore, the present model provides an end-to-end solution, allowing for rapid computation of convergent series for Stokes waves with arbitrary height in arbitrary depth, thus facilitating applications in ocean engineering. In this way, we propose a continuous semi-analytical series description of the Stokes wave family, which shows that periodic waves in deep and intermediate water, cnoidal waves in shallow water, and solitary waves in extremely shallow water can be interpreted within a unified solution framework.