Hostname: page-component-68c7f8b79f-gnk9b Total loading time: 0 Render date: 2025-12-18T08:59:04.224Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  16 December 2025

Chong Lin  and
Shijun Liao Open the ORCID record for Shijun Liao [Opens in a new window]
Chong Lin
Affiliation:
School of Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, PR China
Shijun Liao*
Affiliation:
School of Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, PR China State Key Laboratory of Ocean Engineering, Shanghai 200240, PR China
*
Corresponding author: Shijun Liao, sjliao@sjtu.edu.cn
Get access Rights & Permissions [Opens in a new window]

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.

JFM classification

Waves/Free-surface Flows: Surface gravity waves

Information

Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 1025 , 25 December 2025 , A23
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Brecht, R., Cardoso-Bihlo, E. & Bihlo, A. 2025 Physics-informed neural networks for tsunami inundation modeling. J. Comput. Phys. 114066.10.1016/j.jcp.2025.114066CrossRefGoogle Scholar
Brunton, S.L., Noack, B.R. & Koumoutsakos, P. 2020 Machine learning for fluid mechanics. Annu. Rev. Fluid Mech. 52 (1), 477–508.10.1146/annurev-fluid-010719-060214CrossRefGoogle Scholar
Chappelear, J.E. 1961 Direct numerical calculation of wave properties. J. Geophys. Res. 66 (2), 501–508.10.1029/JZ066i002p00501CrossRefGoogle Scholar
Chen, D., Liao, S. & Xie, B. 2026 An accurate THINC scheme for interface capturing based on homotopy analysis method. J. Comput. Phys. 544, 114420.10.1016/j.jcp.2025.114420CrossRefGoogle Scholar
Chen, L., Li, B., Luo, C. & Lei, X. 2024 Wavenets: physics-informed neural networks for full-field recovery of rotational flow beneath large-amplitude periodic water waves. Engine. Comput. 40 (5), 2819–2839.CrossRefGoogle Scholar
Chen, Y.-Y. & Chang, H.-K. 2024 Accuracy examination of the fourier series approximation for almost limiting gravity waves on deep water. Math. Comput. Appl. 29 (1), 5.CrossRefGoogle Scholar
Clamond, D. & Dutykh, D. 2013 Fast accurate computation of the fully nonlinear solitary surface gravity waves. Comput. Fluids 84, 35–38.10.1016/j.compfluid.2013.05.010CrossRefGoogle Scholar
Clamond, D. & Dutykh, D. 2018 Accurate fast computation of steady two-dimensional surface gravity waves in arbitrary depth. J. Fluid Mech. 844, 491–518.10.1017/jfm.2018.208CrossRefGoogle Scholar
Clamond, D., Dutykh, D. & Durán, A. 2015 A plethora of generalised solitary gravity–capillary water waves. J. Fluid Mech. 784, 664–680.10.1017/jfm.2015.616CrossRefGoogle Scholar
Crew, S.C. & Trinh, P.H. 2016 New singularities for stokes waves. J. Fluid Mech. 798, 256–283.CrossRefGoogle Scholar
Dallaston, M.C. & Mccue, S.W. 2010 Accurate series solutions for gravity-driven stokes waves. Phys. Fluids 22 (8).10.1063/1.3480394CrossRefGoogle Scholar
Dean, R.G. 1965 Stream function representation of nonlinear ocean waves. J. Geophys. Res. 70 (18), 4561–4572.10.1029/JZ070i018p04561CrossRefGoogle Scholar
Dutykh, D. & Clamond, D. 2014 Efficient computation of steady solitary gravity waves. Wave Motion 51 (1), 86–99.10.1016/j.wavemoti.2013.06.007CrossRefGoogle Scholar
Dutykh, D., Clamond, D. & Durán, A. 2016 Efficient computation of capillary–gravity generalised solitary waves. Wave Motion 65, 1–16.CrossRefGoogle Scholar
Dyachenko, S.A., Lushnikov, P.M. & Korotkevich, A.O. 2016 Branch cuts of stokes wave on deep water. Part i: Numerical solution and padé approximation. Stud. Appl. Math. 137 (4), 419–472.10.1111/sapm.12128CrossRefGoogle Scholar
Eeltink, D., Branger, H., Luneau, C., He, Y., Chabchoub, A., Kasparian, J., Van Den Bremer, T.S. & Sapsis, T.P. 2022 Nonlinear wave evolution with data-driven breaking. Nat. Commun. 13 (1), 2343.10.1038/s41467-022-30025-zCrossRefGoogle ScholarPubMed
Fang, H., Tang, L. & Lin, P. 2023 Homotopy analysis of wave transformation over permeable seabeds and porous structures. Ocean Eng. 274, 114087.10.1016/j.oceaneng.2023.114087CrossRefGoogle Scholar
Fenton, J.D. 1985 A fifth-order stokes theory for steady waves. J. Waterway Port Coastal Ocean Eng. 111 (2), 216–234.10.1061/(ASCE)0733-950X(1985)111:2(216)CrossRefGoogle Scholar
Fenton, J.D. 1988 The numerical solution of steady water wave problems. Comput. Geosci. 14 (3), 357–368.10.1016/0098-3004(88)90066-0CrossRefGoogle Scholar
Grant, M.A. 1973 The singularity at the crest of a finite amplitude progressive stokes wave. J. Fluid Mech. 59 (2), 257–262.10.1017/S0022112073001552CrossRefGoogle Scholar
Jouybari, M.A., Yuan, J., Brereton, G.J. & Murillo, M.S. 2021 Data-driven prediction of the equivalent sand-grain height in rough-wall turbulent flows. J. Fluid Mech. 912, A8.Google Scholar
Kolmogorov, A.N. 1991 The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers. Proc. Royal Soc. London Series A: Math. Phys. Sci. 434 (1890), 9–13.Google Scholar
Kutz, J.N. 2017 Deep learning in fluid dynamics. J. Fluid Mech. 814, 1–4.10.1017/jfm.2016.803CrossRefGoogle Scholar
LeCun, Y., Bengio, Y. & Hinton, G. 2015 Deep learning. Nature 521 (7553), 436–444.10.1038/nature14539CrossRefGoogle ScholarPubMed
Li, J., Liu, Z., Borthwick, A.G.L., Cui, J. & Liao, S. 2024 Steady-state interfacial gravity waves with one-dimensional class-iii triad resonance. Phys. Rev. Fluids 9 (9), 094801.10.1103/PhysRevFluids.9.094801CrossRefGoogle Scholar
Li, S., Li, W. & Noack, B.R. 2022 Machine-learned control-oriented flow estimation for multi-actuator multi-sensor systems exemplified for the fluidic pinball. J. Fluid Mech. 952, A36.CrossRefGoogle Scholar
Liao, S. 1992 The proposed homotopy analysis technique for the solution of nonlinear problems. PhD thesis, Shanghai Jiao Tong University,Google Scholar
Liao, S. 1999 A uniformly valid analytic solution of two-dimensional viscous flow over a semi-infinite flat plate. J. Fluid Mech. 385, 101–128.10.1017/S0022112099004292CrossRefGoogle Scholar
Liao, S. 2003 Beyond Perturbation: Introduction to the Homotopy Analysis Method. Chapman and Hall/CRC.10.1201/9780203491164CrossRefGoogle Scholar
Liao, S. 2009 Notes on the homotopy analysis method: some definitions and theorems. Commun. Nonlinear Sci. 14 (4), 983–997.10.1016/j.cnsns.2008.04.013CrossRefGoogle Scholar
Liao, S. 2010 An optimal homotopy-analysis approach for strongly nonlinear differential equations. Commun. Nonlinear Sci. 15 (8), 2003–2016.10.1016/j.cnsns.2009.09.002CrossRefGoogle Scholar
Liao, S. 2011 On the homotopy multiple-variable method and its applications in the interactions of nonlinear gravity waves. Commun. Nonlinear Sci. 16 (3), 1274–1303.10.1016/j.cnsns.2010.06.026CrossRefGoogle Scholar
Liao, S. 2012 Homotopy Analysis Method in Nonlinear Differential Equations. Springer.10.1007/978-3-642-25132-0CrossRefGoogle Scholar
Liao, S. 2023 Avoiding small denominator problems by means of the homotopy analysis method. Adv. Appl. Math. Mech. 15 (2), 267–299.CrossRefGoogle Scholar
Liao, S. 2026 HAM-Schrödingerisation: a generic framework of quantum simulation for any nonlinear PDEs. Adv. Appl. Math. Mech. 18 (1), 1–10.10.4208/aamm.OA-2024-0242CrossRefGoogle Scholar
Liao, S. & Cheung, K.F. 2003 Homotopy analysis of nonlinear progressive waves in deep water. J. Eng. Math. 45, 105–116.10.1023/A:1022189509293CrossRefGoogle Scholar
Ling, J., Kurzawski, A. & Templeton, J. 2016 Reynolds averaged turbulence modelling using deep neural networks with embedded invariance. J. Fluid Mech. 807, 155–166.10.1017/jfm.2016.615CrossRefGoogle Scholar
Liu, Z. & Liao, S. 2014 Steady-state resonance of multiple wave interactions in deep water. J. Fluid Mech. 742, 664–700.10.1017/jfm.2014.2CrossRefGoogle Scholar
Liu, Z., Xu, D.L., Li, J., Peng, T., Alsaedi, A. & Liao, S. 2015 On the existence of steady-state resonant waves in experiments. J. Fluid Mech. 763, 1–23.10.1017/jfm.2014.658CrossRefGoogle Scholar
Longuet-Higgins, H.C. 1975 Integral properties of periodic gravity waves of finite amplitude. Proc. Royal Soc. London A. Mathe. Phys. Sci. 342(,1629), 157–174.Google Scholar
Longuet-Higgins, M.S. & Fox, M.J.H. 1978 Theory of the almost-highest wave. Part 2. Matching and analytic extension. J. Fluid Mech. 85 (4), 769–786.CrossRefGoogle Scholar
Longuet-Higgins, M.S. & Fenton, J.D. 1974 On the mass, momentum, energy and circulation of a solitary wave. ii. Proc. Royal Soc. London A. Math. Phys. Sci. 340 (1623), 471–493.Google Scholar
Lozano-Durán, A. & Bae, H.J. 2023 Machine learning building-block-flow wall model for large-eddy simulation. J. Fluid Mech. 963, A35.10.1017/jfm.2023.331CrossRefGoogle Scholar
Lu, H., Wang, Q., Tang, W. & Liu, H. 2024 Physics-informed neural networks for fully non-linear free surface wave propagation. Phys. Fluids 36 (6), 062106.10.1063/5.0210591CrossRefGoogle Scholar
Lukomsky, V.’, Gandzha, I. & Lukomsky, D. 2002 Steep sharp-crested gravity waves on deep water. Phys. Rev. Lett. 89 (16), 164502.CrossRefGoogle ScholarPubMed
Lushnikov, P.M. 2016 Structure and location of branch point singularities for stokes waves on deep water. J. Fluid Mech. 800, 557–594.10.1017/jfm.2016.405CrossRefGoogle Scholar
Lushnikov, P.M., Dyachenko, S.A. & Silantyev, A.D. 2017 New conformal mapping for adaptive resolving of the complex singularities of stokes wave. Proc. Royal Soc. A: Math. Phys. Eng. Sci. 473 (2202), 20170198.10.1098/rspa.2017.0198CrossRefGoogle ScholarPubMed
Maklakov, D.V. 2002 Almost-highest gravity waves on water of finite depth. Eur. J. Appl. Math. 13 (1), 67–93.CrossRefGoogle Scholar
Raissi, M., Perdikaris, P. & Karniadakis, G.E. 2019 Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. J. Comput. Phys. 378, 686–707.10.1016/j.jcp.2018.10.045CrossRefGoogle Scholar
Rienecker, M.M. & Fenton, J.D. 1981 A fourier approximation method for steady water waves. J. Fluid Mech. 104, 119–137.10.1017/S0022112081002851CrossRefGoogle Scholar
Schwartz, L.W. 1974 Computer extension and analytic continuation of stokes’ expansion for gravity waves. J. Fluid Mech. 62 (3), 553–578.10.1017/S0022112074000802CrossRefGoogle Scholar
Stokes, G.G. 1847 On the theory of oscillatory waves. Trans. Cambridge Philosop. Soc. 8, 441–455.Google Scholar
Stokes, G.G. 1880 Supplement to a paper on the theory of oscillatory waves. Math. Phys. Papers 1, 314–326.Google Scholar
Tanveer, S. 1991 Singularities in water waves and rayleigh–taylor instability. Proc. Royal Soc. London Series A: Math Phys. Sci. 435 (1893), 137–158.Google Scholar
Vajravelu, K. & Van Gorder, R. 2013 Nonlinear Flow Phenomena and Homotopy Analysis, vol. 2. Springer.Google Scholar
Van, G., Robert, A. & Vajravelu, K. 2008 Analytic and numerical solutions to the lane–emden equation. Phys. Lett. A 372 (39), 6060–6065.Google Scholar
Wang, X. & Jiang, H. 2024 Physics-guided deep learning for skillful wind-wave modeling. Sci. Adv. 10 (49), eadr3559.10.1126/sciadv.adr3559CrossRefGoogle ScholarPubMed
Williams, J.M. 1981 Limiting gravity waves in water of finite depth. Philosop. Trans. Royal Soc. London Series A Math. Phys. Sci. 302(,1466), 139–188.Google Scholar
Wilton, J.R. 1914 On deep water waves. London Edinburgh Dublin Philosop. Mag. J. Sci. 27 (158), 385–394.10.1080/14786440208635100CrossRefGoogle Scholar
Xie, C., Yuan, Z. & Wang, J. 2020 Artificial neural network-based nonlinear algebraic models for large eddy simulation of turbulence. Phys. Fluids 32 (11).10.1063/5.0025138CrossRefGoogle Scholar
Xu, D., Lin, Z., Liao, S. & Stiassnie, M. 2012 On the steady-state fully resonant progressive waves in water of finite depth. J. Fluid Mech. 710, 379–418.10.1017/jfm.2012.370CrossRefGoogle Scholar
Xu, D., Wang, J., Yu, C. & Chen, S. 2023 Artificial-neural-network-based nonlinear algebraic models for large-eddy simulation of compressible wall-bounded turbulence. J. Fluid Mech. 960, A4.10.1017/jfm.2023.179CrossRefGoogle Scholar
Yousefi, K., Hora, G.S., Yang, H., Veron, F. & Giometto, M.G. 2024 A machine learning model for reconstructing skin-friction drag over ocean surface waves. J. Fluid Mech. 983, A9.10.1017/jfm.2024.81CrossRefGoogle Scholar
Zhang, D., Chen, Y. & Chen, S. 2024 Filtered partial differential equations: a robust surrogate constraint in physics-informed deep learning framework. J. Fluid Mech. 999, A40.10.1017/jfm.2024.471CrossRefGoogle Scholar
Zhao, K. & Liu, P.L.-F. 2022 On stokes wave solutions. Proc. Royal Soc. A 478 (2258), 20210732.CrossRefGoogle Scholar
Zhong, X. & Liao, S. 2017 Analytic solutions of von kármán plate under arbitrary uniform pressure – equations in differential form. Stud. Appl. Math. 138 (4), 371–400.CrossRefGoogle Scholar
Zhong, X. & Liao, S. 2018 On the limiting stokes wave of extreme height in arbitrary water depth. J. Fluid Mech. 843, 653–679.10.1017/jfm.2018.171CrossRefGoogle Scholar
Zhu, L., Jiang, X., Lefauve, A., Kerswell, R.R. & Linden, P.F. 2024 New insights into experimental stratified flows obtained through physics-informed neural networks. J. Fluid Mech. 981, R1.10.1017/jfm.2024.49CrossRefGoogle Scholar