Rui Cao

Because high-resolution spatial wave measurements are difficult to obtain in the field, individual-wave geometry is often estimated from point-based time series, with wavelength inferred by applying the linear dispersion relation to the measured wave period. We evaluate the accuracy of this approach using time-synchronised digital cameras to record the temporal and spatial evolution of unidirectional, non-breaking waves in laboratory-generated random sea states. Individual waves are identified through zero-crossing analysis and matched between the time and space domains using the digital imagery. In total, we have analysed almost 55,000 matched waves encompassing both unforced and wind-forced conditions. For each wave, direct spatial measurements of crest height, wavelength, and steepness are compared with the corresponding values derived from point-based time-domain records. The results show that time-and space-domain crest heights agree closely, as expected, but time-derived wavelength and wave steepness estimates show larger discrepancies from their spatially measured values. A wave-type classification shows that these differences are most pronounced for waves containing additional local peaks within their spatial wavelength or temporal period, for which we use the term ”irregular”, characteristic of shorter waves superimposed on longer waves. Temporally-derived and spatially-measured wavelength and steepness are typically in good agreement for ”regular-like” waves across the sea states examined. We further introduce a paired zero-crossing wave-combination scheme that identifies waves with more reliable time-derived geometric estimates while retaining approximately 80% of the observations. These findings provide a practical approach for improving estimates of individual-wave geometry when only point-based temporal measurements are available. Plain Language Summary Ocean waves are usually measured at a fixed point using buoys or gauges. To estimate wavelength and wave steepness that describe the wave shape in space, the linear dispersion relation is used. However, this conversion relies on assumptions that may not hold for unsteady waves. We used high-resolution cameras in a laboratory wave channel to measure nearly 55,000 individual waves in both time and space, allowing us to directly compare wave properties measured in space with those estimated from time records. While wave crest heights match very well, wavelength and steepness can differ significantly, especially when wind is present. We therefore developed a scheme that combines two standard methods of separating waves in time to identify a large subset of waves with more reliable geometric estimates. These findings can improve the interpretation of point-based wave records used in ocean science and engineering.

Data-driven discovery of partial differential equations (PDEs) is a promising approach for uncovering the underlying laws governing complex systems. However, purely data-driven techniques face the dilemma of balancing search space with optimization efficiency. This study introduces a knowledge-guided approach that incorporates existing PDEs documented in a mathematical handbook to facilitate the discovery process. These PDEs are encoded as sentence-like structures composed of operators and basic terms, and used to train a generative model, called EqGPT, which enables the generation of free-form PDEs. A loop of “generation–evaluation–optimization” is constructed to autonomously identify the most suitable PDE. Experimental results demonstrate that this framework can recover a variety of PDE forms with high accuracy and computational efficiency, particularly in cases involving complex temporal derivatives or intricate spatial terms, which are often beyond the reach of conventional methods. The approach also exhibits generalizability to irregular spatial domains and higher dimensional settings. Notably, it succeeds in discovering a previously unreported PDE governing strongly nonlinear surface gravity waves propagating toward breaking, based on real-world experimental data, highlighting its applicability to practical scenarios and its potential to support scientific discovery.

Many supervised machine learning methods have revolutionised the empirical modelling of complex systems. These empirical models, however, are usually "black boxes" and provide only limited physical explanations about the underlying systems. Instead, so-called "knowledge discovery" methods can be used to explore the governing equations that describe observed phenomena. This paper focuses on how we can use such methods to explore underlying physics and also model a commonly observed yet not fully understood phenomenon - the breaking of ocean waves. In our work, we use symbolic regression to explore the equation that describes wave-breaking evolution from a dataset of in silico waves generated using expensive numerical methods. Our work discovers a new boundary equation that provides a reduced-order description of how the surface elevation (i.e., the water-air interface) evolves forward in time, including the instances when the wave breaks - a problem that has defied traditional approaches. Compared to the existing empirical models, the unique equation-based nature of our model allows further mathematical interpretation, which provides an opportunity to explore the fundamentals of breaking waves. Further expert-AI collaborative research reveals the physical meaning of each term of the discovered equation, which suggests a new characteristic of breaking waves in deep water - a decoupling between the water-air interface and the fluid velocities. This novel reduced-order model also hints at computationally efficient ways to simulate breaking waves for engineering applications.