From Video-to-PDE: Data-Driven Discovery of Nonlinear Dye Plume Dynamics
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arXiv
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| Format: | Preprint |
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2026
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| author | Acosta-Minoli, Cesar Sarkar, Sayantan |
| author_facet | Acosta-Minoli, Cesar Sarkar, Sayantan |
| contents | 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 $u_t+\mathbf v(t)\!\cdot\!\nabla u = 9.005\,|\nabla u|^{2}+0.666\,Δu$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_04535 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | From Video-to-PDE: Data-Driven Discovery of Nonlinear Dye Plume Dynamics Acosta-Minoli, Cesar Sarkar, Sayantan Machine Learning Numerical Analysis Computational Physics Applications 35R30, 68T07, 65M32, 35K55 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 $u_t+\mathbf v(t)\!\cdot\!\nabla u = 9.005\,|\nabla u|^{2}+0.666\,Δu$ 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. |
| title | From Video-to-PDE: Data-Driven Discovery of Nonlinear Dye Plume Dynamics |
| topic | Machine Learning Numerical Analysis Computational Physics Applications 35R30, 68T07, 65M32, 35K55 |
| url | https://arxiv.org/abs/2605.04535 |