- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review Fluids

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Mapping surface height dynamics to subsurface flow physics in free-surface turbulent flow using a shallow recurrent decoder

Kristoffer S. Moen1,*, Jørgen R. Aarnes1,*, Simen Å. Ellingsen1, and J. Nathan Kutz2,3

  • *K.S.M. and J.R.A. are to be considered joint first authors.
PDF

Phys. Rev. Fluids 11, 084601 – Published 3 August, 2026

DOI: https://doi.org/10.1103/3q12-ylb6

Abstract

Near-surface turbulent flows beneath a free surface are reconstructed from sparse measurements of the surface height variation, by a neural network algorithm known as the SHallow REcurrent Decoder (SHRED). The reconstruction of turbulent flow fields from limited, partial, or indirect measurements remains a grand challenge in science and engineering. The central goal in such applications is to leverage easy-to-measure proxy variables in order to estimate quantities which have not been, and perhaps cannot in practice be, measured. In the application considered here, the aim is to use a sparse number of surface height point measurements of a flow field, or drone video footage of surface features, in order to infer the turbulent flow field beneath the surface. SHRED is a deep learning architecture that learns a delay-coordinate embedding from a few surface height (point) sensors and maps it, via a shallow decoder trained in a compressed basis, to full subsurface fields, enabling fast, robust training from minimal data. We demonstrate the SHRED sensing architecture on two types of turbulent data from recent studies [Aarnes et al. J. Fluid Mech. 1007, A38 (2025) and Babiker et al. Phys. Rev. Fluids 11, 054802 (2026), respectively]: fully resolved DNS data and PIV laboratory data from a turbulent water tank. SHRED is capable of robustly mapping surface height fluctuations to full-state flow fields up to about one integral length scale deep, with as few as three surface measurements.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (67)

  1. M. Brocchini and D. H. Peregrine, The dynamics of strong turbulence at free surfaces. Part 1. Description, J. Fluid Mech. 449, 225 (2001).
  2. J. R. Aarnes, O. M. Babiker, A. Xuan, L. Shen, and S. Å. Ellingsen, Vortex structures under dimples and scars in turbulent free-surface flows, J. Fluid Mech. 1007, A38 (2025).
  3. O. M. Babiker, I. Bjerkebæk, A. Xuan, L. Shen, and S. Å. Ellingsen, Vortex imprints on a free surface as proxy for surface divergence, J. Fluid Mech. 964, R2 (2023).
  4. E. A. D'Asaro, Turbulence in the upper-ocean mixed layer, Ann. Rev. Marine Sci. 6, 101 (2014).
  5. R. Wanninkhof, W. E. Asher, D. T. Ho, C. Sweeney, and W. R. McGillis, Advances in quantifying air-sea gas exchange and environmental forcing, Annu. Rev. Mar. Sci. 1, 213 (2009).
  6. C. B. Brinkerhoff, C. J. Gleason, C. J. Zappa, P. A. Raymond, and M. E. Harlan, Remotely sensing river greenhouse gas exchange velocity using the SWOT satellite, Global Biogeochem. Cycles 36, e2022GB007419 (2022).
  7. G. Dolcetti, B. Hortobágyi, M. Perks, S. J. Tait, and N. Dervilis, Using noncontact measurement of water surface dynamics to estimate river discharge, Water Resour. Res. 58, e2022WR032829 (2022).
  8. N. B. Erichson, L. Mathelin, Z. Yao, S. L. Brunton, M. W. Mahoney, and J. N. Kutz, Shallow neural networks for fluid flow reconstruction with limited sensors, Proc. R. Soc. A 476, 20200097 (2020).
  9. J. Zhang, J. Liu, and Z. Huang, Improved deep learning method for accurate flow field reconstruction from sparse data, Ocean Eng. 280, 114902 (2023).
  10. P. Pant and A. B. Farimani, Deep learning for efficient reconstruction of high-resolution turbulent DNS data, arXiv:2010.11348.
  11. Z. Deng, C. He, Y. Liu, and K. C. Kim, Super-resolution reconstruction of turbulent velocity fields using a generative adversarial network-based artificial intelligence framework, Phys. Fluids 31, 125111 (2019).
  12. S. Discetti and Y. Liu, Machine learning for flow field measurements: A perspective, Meas. Sci. Technol. 34, 021001 (2022).
  13. Á. Moreno Soto, A. Güemes, and S. Discetti, Complete flow characterization from snapshot PIV, fast probes and physics-informed neural networks, Comput. Methods Appl. Mech. Eng. 419, 116652 (2024).
  14. X. Jin, S. Laima, W.-L. Chen, and H. Li, Time-resolved reconstruction of flow field around a circular cylinder by recurrent neural networks based on non-time-resolved particle image velocimetry measurements, Exp. Fluids 61, 114 (2020).
  15. Z. Deng, Y. Chen, Y. Liu, and K. C. Kim, Time-resolved turbulent velocity field reconstruction using a long short-term memory (LSTM)-based artificial intelligence framework, Phys. Fluids 31, 075108 (2019).
  16. G. Borrelli, L. Guastoni, H. Eivazi, P. Schlatter, and R. Vinuesa, Predicting the temporal dynamics of turbulent channels through deep learning, Int. J. Heat Fluid Flow 96, 109010 (2022).
  17. S. L. Brunton and J. N. Kutz, Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control (Cambridge University Press, Cambridge, 2019).
  18. P. A. Srinivasan, L. Guastoni, H. Azizpour, P. Schlatter, and R. Vinuesa, Predictions of turbulent shear flows using deep neural networks, Phys. Rev. Fluids 4, 054603 (2019).
  19. S. L. Brunton, B. R. Noack, and P. Koumoutsakos, Machine learning for fluid mechanics, Annu. Rev. Fluid Mech. 52, 477 (2020).
  20. A. Cuéllar, A. Güemes, A. Ianiro, Ó. Flores, R. Vinuesa, and S. Discetti, Three-dimensional generative adversarial networks for turbulent flow estimation from wall measurements, J. Fluid Mech. 991, A1 (2024).
  21. M. Milano and P. Koumoutsakos, Neural network modeling for near wall turbulent flow, J. Comput. Phys. 182, 1 (2002).
  22. A. Xuan and L. Shen, Reconstruction of three-dimensional turbulent flow structures using surface measurements for free-surface flows based on a convolutional neural network, J. Fluid Mech. 959, A34 (2023).
  23. G. S. Hora, P. Gentine, M. Momen, and M. G. Giometto, Physics-informed data-driven reconstruction of turbulent wall-bounded flows from planar measurements, Phys. Fluids 36, 115191 (2024).
  24. K. Poulinakis, D. Drikakis, I. W. Kokkinakis, S. M. Spottswood, and T. Dbouk, LSTM reconstruction of turbulent pressure fluctuation signals, Computation 12, 4 (2024).
  25. S. Gakhar, J. R. Koseff, and N. T. Ouellette, Extracting free-surface expressions of underwater features, Exp. Fluids 63, 138 (2022).
  26. L. Guastoni, A. Güemes, A. Ianiro, S. Discetti, P. Schlatter, H. Azizpour, and R. Vinuesa, Convolutional-network models to predict wall-bounded turbulence from wall quantities, J. Fluid Mech. 928, A27 (2021).
  27. A. Güemes, S. Discetti, and A. Ianiro, Sensing the turbulent large-scale motions with their wall signature, Phys. Fluids 31, 125112 (2019).
  28. J. P. Williams, O. Zahn, and J. N. Kutz, Sensing with shallow recurrent decoder networks, Proc. R. Soc. A 480, 20240054 (2024).
  29. F. Faraji, M. Reza, and J. N. Kutz, Shallow recurrent decoder for reduced order modeling of E×B plasma dynamics, Mach. Learn.: Sci. Technol. 6, 025024 (2025).
  30. M. Tomasetto, J. P. Williams, F. Braghin, A. Manzoni, and J. N. Kutz, Reduced order modeling with shallow recurrent decoder networks, Nat. Commun. 16, 10260 (2025).
  31. F. Takens, Detecting strange attractors in turbulence, in Dynamical Systems and Turbulence, edited by D. A. Rand and L.-S. Young, Lecture Notes in Mathematics (Springer-Verlag, 1981), Vol. 898, pp. 366–381.
  32. S. Riva, C. Introini, A. Cammi, and J. N. Kutz, Robust state estimation from partial out-core measurements with shallow recurrent decoder for nuclear reactors, Prog. Nucl. Energy 189, 105928 (2025).
  33. C. Introini, S. Riva, J. N. Kutz, and A. Cammi, From models to experiments: Shallow recurrent decoder networks on the DYNASTY experimental facility, arXiv:2503.08907.
  34. M. R. Ebers, J. P. Williams, K. M. Steele, and J. N. Kutz, Leveraging arbitrary mobile sensor trajectories with shallow recurrent decoder networks for full-state reconstruction, IEEE Access 12, 97428 (2024).
  35. M. L. Gao, J. P. Williams, and J. N. Kutz, Sparse identification of nonlinear dynamics and Koopman operators with shallow recurrent decoder networks, Proc. Natl. Acad. Sci. USA 123, e2508144123 (2026).
  36. J. R. Aarnes, O. M. Babiker, A. Xuan, L. Shen, and S. Å Ellingsen, Replication data for: Vortex structures under dimples and scars in turbulent free-surface flows (2025), https://doi.org/10.18710/XQ81WH.
  37. O. M. Babiker, J. R. Aarnes, A. Semati, A. Ferran, Y. H. Tee, R. J. Hearst, and S. Å. Ellingsen, Replication data for: Experimental investigation relating free-surface features to sub-surface turbulence (2026), https://doi.org/10.18710/TSIZCQ.
  38. O. M. Babiker, J. R. Aarnes, A. Semati, A. Ferran, Y. H. Tee, R. J. Hearst, and S. Å. Ellingsen, Experimental investigation relating free-surface features to subsurface turbulence, Phys. Rev. Fluids 11, 054802 (2026).
  39. B. H. Brumley and G. H. Jirka, Near-surface turbulence in a grid-stirred tank, J. Fluid Mech. 183, 235 (1987).
  40. L. Chiapponi, S. Longo, and M. Tonelli, Experimental study on oscillating grid turbulence and free surface fluctuation, Exp. Fluids 53, 1515 (2012).
  41. H. Herlina and G. H. Jirka, Experiments on gas transfer at the air–water interface induced by oscillating grid turbulence, J. Fluid Mech. 594, 183 (2008).
  42. E. J. Hopfinger and J. A. Toly, Spatially decaying turbulence and its relation to mixing across density interfaces, J. Fluid Mech. 78, 155 (1976).
  43. T. Jamin, M. Berhanu, and E. Falcon, Experimental study of three-dimensional turbulence under a free surface, Phys. Rev. Fluids 10, 034608 (2025).
  44. T. Lacassagne, M. El-Hajem, F. Morge, S. Simoens, and J.-Y. Champagne, Study of gas liquid mass transfer in a grid stirred tank, Oil Gas Sci. Tech.–Revue d'IFP Energies Nouvelles 72, 7 (2017).
  45. S. P. McKenna and W. R. McGillis, The role of free-surface turbulence and surfactants in air–water gas transfer, Int. J. Heat Mass Transf. 47, 539 (2004).
  46. S. M. Thompson and J. S. Turner, Mixing across an interface due to turbulence generated by an oscillating grid, J. Fluid Mech. 67, 349 (1975).
  47. E. A. Variano and E. A. Cowen, Turbulent transport of a high-Schmidt-number scalar near an air–water interface, J. Fluid Mech. 731, 259 (2013).
  48. X. Guo and L. Shen, Interaction of a deformable free surface with statistically steady homogeneous turbulence, J. Fluid Mech. 658, 33 (2010).
  49. H. Herlina and J. G. Wissink, Direct numerical simulation of turbulent scalar transport across a flat surface, J. Fluid Mech. 744, 217 (2014).
  50. H. Herlina and J. G. Wissink, Simulation of air–water interfacial mass transfer driven by high-intensity isotropic turbulence, J. Fluid Mech. 860, 419 (2019).
  51. G. B. Folland, Introduction to Partial Differential Equations, 2nd ed. (Princeton University Press, Princeton, 1995).
  52. J. N. Kutz, Data-Driven Modeling & Scientific Computation: Methods for Complex Systems & Big Data (Oxford University Press, 2013).
  53. J. N. Kutz, M. Reza, F. Faraji, and A. Knoll, Shallow recurrent decoder for reduced order modeling of plasma dynamics, arXiv:2405.11955.
  54. https://github.com/jaarnes/fst_SHRED.
  55. X. Guo and L. Shen, On the generation and maintenance of waves and turbulence in simulations of free-surface turbulence, J. Comput. Phys. 228, 7313 (2009).
  56. A. Xuan and L. Shen, A conservative scheme for simulation of free-surface turbulent and wave flows, J. Comput. Phys. 378, 18 (2019).
  57. A. Xuan and L. Shen, Analyses of wave-phase variation of Reynolds shear stress underneath surface wave using streamline coordinates, J. Fluid Mech. 931, A32 (2022).
  58. A. Semati, A. Shankaran, B. K. Smeltzer, E. Æsøy, R. J. Hearst, and S. Å. Ellingsen, Simultaneous free-surface profilometry and subsurface velocimetry with fringe projection and PIV, Exp. Fluids 67, 120 (2026).
  59. D. J. Ruth and F. Coletti, Structure and energy transfer in homogeneous turbulence below a free surface, J. Fluid Mech. 1001, A46 (2024).
  60. T. S. Lundgren, Linearly forced isotropic turbulence, Center for Turbulence Research Annual Research Briefs, 2023, https://web.stanford.edu/group/ctr/ResBriefs03/lundgren.pdf.
  61. C. Rosales and C. Meneveau, Linear forcing in numerical simulations of isotropic turbulence: Physical space implementations and convergence properties, Phys. Fluids 17, 095106 (2005).
  62. G. Strang, Linear Algebra and Learning from Data (Cambridge Press, Wellesley, MA, 2019).
  63. Z. Wang and A. C. Bovik, Mean squared error: Love it or leave it? A new look at signal fidelity measures, IEEE Sign. Proc. Mag. 26, 98 (2009).
  64. Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, Image quality assessment: From error visibility to structural similarity, IEEE Trans. on Image Process. 13, 600 (2004).
  65. L. Shen, X. Zhang, D. K. P. Yue, and G. S. Triantafyllou, The surface layer for free-surface turbulent flows, J. Fluid Mech. 386, 167 (1999).
  66. S. Chen, T. Bao, P. Givi, C. Zheng, and X. Jia, Reconstructing turbulent flows using spatio-temporal physical dynamics, ACM Trans. Intell. Syst. Technol. 15, 1 (2024).
  67. X. Zhang, Y. Bao, M. L. Gao, and J. N. Kutz, Sendai: A hierarchical sparse-measurement, efficient data assimilation framework, arXiv:2601.21664.

Outline

Information

Phys. Rev. Fluids 11, 084601– Published 3 August, 2026
  • Received 7 October 2025
  • Accepted 7 July 2026
Article part of CHORUS
Crossmark - Check for updates button

©2026 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation