Rediscovering shallow water equations from experimental data

TL;DR

The paper tackles the problem of discovering governing PDEs from experimental observations using inexpensive, everyday equipment. It develops and compares two distinct data-driven approaches—WSINDy (weak-form sparse regression) and a novel Fourier-multiplier method—to identify a KdV-type PDE from shallow-water soliton videos, applying them to a benchtop flume dataset. Both methods converge on the same evolution equation, ∂_t H = c_{1,1} ∂_x H + c_{3,1} ∂^3_x H + c_{5,1} ∂^5_x H + c_{1,2} ∂_x(H^2), and forward-simulations using the learned coefficients exhibit accurate predictive skill on withheld solitons, with cumulative errors well below 20% of the amplitude. The work demonstrates that data from everyday experiments can robustly reveal governing dynamics, and it highlights the fifth-order term as a measurable feature, suggesting GoPro-scale physics as a practical platform for benchmarking new equation-discovery methods.

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.

Figures (11)

• Figure 1: (i) Example frame from recorded video, (ii) illustration of the edge detection algorithm, (iii) full $(x,t)$ field.
• Figure 2: (a) Shows the regression error defined by \ref{['eq:SINDy_regression_error']} for ${M=100}$ and ${K=1000}$. (b) Shows the mean error in real space for the same coefficients as in (a). In (c), the errors are multiplied together, and the star shows the choice of domain size for the results after this.
• Figure 3: The upper row shows the conditional mean magnitude and variance of the coefficients $c_{\mu,\lambda}$ for $M=1000$ ensembles and ${N_x \times N_t = 200\times30}$ for different identified PDEs. The second row shows the real-space error and how often that PDE is identified.
• Figure 4: Top row: One high amplitude and one low amplitude soliton before (blue) and after (red) projection onto the used spatial frequencies. Bottom row: The real (green) and imaginary (orange) parts of the spectrum of the same solitons. The part of the spectrum that is used is shaded in gray.
• Figure 5: Left: The discrete linear multiplier (orange dots) and approximating polynomial $p^5_{\mathrm{lin}}$ (blue line). Right: The discrete nonlinear multiplier (orange dots) and approximating polynomial $p^1_{\mathrm{nonlin}}$ (blue line).