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

Near-surface turbulent flows beneath a free surface are reconstructed from sparse measurements of the surface height variation, by a novel neural network algorithm known as the {\em 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. Specifically, 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 \emph{et al.} J.~Fluid Mech.\ $\mathbf{\text{1007}}$ A38, 2025 and Babiker \emph{et al.} arXiv:251003732, 2025, 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 two integral length scales deep, with as few as three surface measurements.