From Video-to-PDE: Data-Driven Discovery of Nonlinear Dye Plume Dynamics

TL;DR

This paper introduces a pipeline that converts uncalibrated video recordings of dye plumes into interpretable PDE models, enabling data-driven discovery of nonlinear transport dynamics with uncertainty quantification.

Contribution

It develops a novel video-to-PDE framework that isolates transport laws from noisy, uncalibrated visual data using sparse regression and physics-informed refinement.

Findings

The reduced model outperforms advection--diffusion baselines on test frames.

The model retains a positive Laplacian coefficient and admits a Cole--Hopf reduction.

Uncertainty is quantified via a bootstrap approach.

Abstract

Inferring continuum models directly from video is hampered by two facts: the recorded field is uncalibrated image intensity rather than a physical state, and direct numerical differentiation of noisy frames is unstable. We develop a video-to-PDE pipeline that converts grayscale recordings of an ink plume into a normalised scalar field $u(x,y,t)$, isolates a bulk drift $\mathbf{v}(t)$ from intrinsic spreading via the intensity-weighted centroid, and identifies an effective transport law by weak-form sparse regression. Conditioning, threshold-sweep and random-centre diagnostics show that overcomplete libraries are strongly collinear; the search is therefore restricted to compact gradient-based libraries. Coefficients are refined by an inverse physics-informed network and recalibrated against forward rollouts, with a chronological block bootstrap quantifying uncertainty. The selected reduced model $u_t+\mathbf v(t)\!\cdot\!\nabla u = 9.005\,|\nabla u|^{2}+0.666\,\Delta u$ outperforms advection--diffusion baselines on held-out frames, retains a positive Laplacian coefficient, and admits a Cole--Hopf reduction to a linear advection--diffusion equation. The framework demonstrates that uncalibrated visual data can yield compact, predictive and structurally interpretable continuum models when discovery, calibration and uncertainty are treated as distinct stages.