Rediscovering shallow water equations from experimental data

TL;DR

This paper demonstrates data-driven discovery of the shallow-water PDE from video recordings of solitons, validating the models with experimental data and comparing two different identification methods.

Contribution

It introduces a robust approach to rediscover PDEs from experimental videos using two novel methods, validating the results with unseen data.

Findings

Both methods recover the same PDE with similar accuracy.

The discovered PDE accurately predicts soliton dynamics in new experiments.

The approaches are robust to method choice and data variations.

Abstract

New data-driven methods have advanced the discovery of governing equations from observations, enabling parsimonious models for complex systems. Here, we 'rediscover' a shallow-water equation closely related to Korteweg--de Vries (KdV) using only video recordings of solitons in a simple flume. Two fundamentally different approaches -- weak-form sparse identification of nonlinear dynamics (WSINDy) and a novel Fourier-multiplier method -- recover the same PDE, demonstrating that the equation is inherent in the data and robust to the choice of method. Both identify the same terms with comparable magnitudes and errors. To validate the models, we solve the discovered equations forward in time and compare them with additional experimental cases that were not used in the discovery. Based on the results, we discuss absolute and cumulative errors, as well as the strengths and limitations of the two discovery approaches. Together, these results demonstrate the potential of equation discovery from everyday experiments ('GoPro physics') and highlight shallow-water waves as an ideal test bed for developing and benchmarking new methods.