Rediscovering shallow water equations from experimental data

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.

• The paper recovers a PDE formulation resembling the KdV equation from soliton propagation data in a controlled shallow-water flume experiment.
• The study applies two independent equation discovery methods—WSINDy and a Fourier-multiplier approach—to robustly extract governing dynamics despite moderate noise.
• The methods yield consistent coefficient estimations and low error metrics, verifying the experimental model’s predictive stability and practical applicability.

Data-Driven Rediscovery of Shallow Water Equations from Soliton Experiments

Introduction

The paper addresses the central challenge of extracting governing PDEs for fluid dynamics from spatiotemporal observations. Focusing on shallow-water wave dynamics, the authors utilize high-resolution video data of soliton propagation in a controlled flume experiment and apply two independent equation discovery methods—Weak-form Sparse Identification of Nonlinear Dynamics (WSINDy) and a Fourier-multiplier-based approach. Both methods robustly recover a PDE formulation akin to the Korteweg–de Vries (KdV) equation directly from the trajectory of the water surface, confirming the dynamics encoded in the acquired data and highlighting methodological convergence.

Experimental Framework and Data Acquisition

The experimental setup consists of a simple laboratory flume equipped for high-speed videography of unidirectional soliton waves. The primary data source is the time-evolving free-surface elevation, which is extracted from the video data using side-view edge detection algorithms (cf. Canny [canny1986computational], Cao et al. [cao2024identification]). This processing yields a spatiotemporal $\eta(x,t)$ field suitable for downstream dynamic identification. The accuracy of this procedure is critical: robust edge detection and sub-pixel interpolation minimize measurement and digitization artefacts, as corroborated by validation against ground-truth metrics.

Sparse Equation Discovery Methods

WSINDy Methodology

WSINDy builds on SINDy [brunton2016discovering] by casting the discovery problem in a weak, Galerkin-style form, addressing numerical instability in derivative estimation from data. This circumvents issues related to noise amplification in finite-difference schemes. The identification pipeline includes:

1. Construction of a wide dictionary of candidate terms (monomials and derivatives).
2. Integration against suitable test functions (e.g., polynomials, wavelets) to produce weak-form coefficients.
3. Use of regularized sparse regression (LASSO/elastic net [tibshirani1996regression, zou2005regularization]) to select the PDE terms.

The resulting regression selects terms expected for shallow-water wave physics: $\eta_t$, $\eta_x$, $\eta_{xxx}$, and nonlinear like $\eta\eta_x$, revealing equivalence with generalized KdV-family PDEs. PySINDy [desilva2020, Kaptanoglu2022] is leveraged for reproducibility and benchmarking.

Fourier-Multiplier Approach

The Fourier-multiplier strategy, inspired by recent advances in symbolic regression for PDEs, projects the data and candidate term dictionary into the Fourier domain and applies regression in spectral space. This method exploits the benefits of global spectral differentiation, reducing boundary artefacts and improving sensitivity to dispersive terms. The Fourier-identified equation matches the WSINDy result across coefficients and structure, and demonstrates consistency even with moderate noise and measurement error, reinforcing the physical legitimacy of the discovered model.

Model Validation and Error Analysis

Forward-integration of the discovered PDEs, using initial conditions from withheld experimental trials, demonstrates predictive fidelity over observable time windows. Quantitative error metrics—absolute error, cumulative error, and time-wise drift—are computed:

• Absolute and cumulative errors remain subpixel for all variables over multiple soliton cycles, substantiating that the model generalizes beyond the training regime.
• No systematic drift is observed, indicating stability in long-time integration (with time-stepping via standard solvers like adaptive Runge–Kutta or Kreiss–Oliger).

Comparative analysis between WSINDy and Fourier-multiplier solutions shows negligible discrepancy. Notably, both approaches correctly estimate the sign and magnitude of key nonlinear and dispersive coefficients, essential for accurate soliton representation.

Methodological Strengths and Limitations

Both approaches are resilient against moderate measurement noise. The weak-form regression is robust to spatially localized artefacts and noisy derivatives, suitable for practical experiment-derived datasets. The Fourier-multiplier method excels at extracting higher-order dispersive and nonlocal terms, valuable when extending equation discovery to systems with significant spectral content. However, both methods are limited by their reliance on well-resolved video data and accurate free-surface tracking; significant deviations in measurement quality degrade discovery performance.

Implications and Future Directions

The results assert the feasibility of extracting canonical PDEs from non-specialized experimental imaging, underpinning a paradigm of "GoPro physics"—equation discovery using consumer-grade instrumentation. The shallow-water flume system, due to its rich soliton dynamics and analytically tractable PDE structures, is validated as an excellent testbed for benchmarking new symbolic regression and sparse identification algorithms.

The generality and convergence of WSINDy and Fourier-multiplier methods suggest potential for extension to other canonical systems in nonlinear science (e.g., Navier–Stokes, reaction–diffusion, capillary–gravity waves [gao2021capillary]). The demonstrated pipeline provides a blueprint for future research directed at:

• Discovering governing equations from more complex, higher-dimensional fluid systems.
• Applying uncertainty quantification and Bayesian approaches [cheng2023] to further improve robustness.
• Extending methodologies to settings with latent dynamics or missing variables [reinbold2019data].
• Developing active learning and experiment design strategies to optimize data acquisition for equation discovery.

Conclusion

The paper establishes robust and generalizable data-driven PDE identification pipelines for shallow-water soliton dynamics, verified via two fundamentally different sparse regression frameworks. The strong agreement between the methodologies and experimental data supports the notion that governing physics can be uncovered with minimal a priori information, leveraging only high-speed video and modern regression tools. These results affirm the viability of experimental equation discovery for real-world systems and lay the groundwork for systematic exploration and benchmarking of future methods in dynamical system identification.