From Video-to-PDE: Data-Driven Discovery of Nonlinear Dye Plume Dynamics
Authors: Cesar Acosta-Minoli, Sayantan Sarkar
Organizations: aGEDES, Universidad del Quindio, Cra 15 Cl 12-00, Armenia, 630004, Quindio, Colombia · bDepartment of Mathematics, State University of New York at Buffalo, Mathematics Building, Buffalo, 14260, New York, USA
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 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 ut+v(t)⋅∇u=9.005∣∇u∣2+0.666Δu 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.
To study the ability to infer physical dynamics from videos and extrapolate them forward in time, we assemble a dataset of 2D Material Point Method (MPM) physical simulations covering rich physical phenomena such as deformable objects, fluids, kinetic objects, and emitters. We study code generation and video diffusion approaches on this dataset, identifying their strengths and weaknesses by varying the amount of physically relevant side information. The code generation model, beyond giving a working demonstration of automatic synthesis of MPM simulations, reveals that such an approach struggles with inferring physical parameters from visual input, but relative to video diffusion, produces physically and temporally stable extrapolations forward in time, while the video diffusion model more strongly identifies geometric properties from visual input but produces physically implausible extrapolations.
Bridging the gap between visual realism and physical understanding is a core challenge for video-based world models. We study the structural identifiability of continuous-time physical laws from raw pixels, focusing on whether an encoder-only pipeline can uniquely recover the parameters of second-order linear ODEs. We prove that a level-set slope-coverage condition ensures the learned latent space is locally affine to the true physical state, enabling exact parameter recovery. Our theory provides the first characterization of minimal data requirements across damping regimes, establishing that underdamped systems are identifiable from a single video clip, whereas other regimes require three diverse trajectories. We further introduce a variance-floor regularizer to stabilize the decoder-free objective and prevent latent collapse. Validated on synthetic and real-world data, our approach demonstrates that interpretable physical constants can be reliably estimated from video without the need for compute-intensive pixel reconstruction, ensuring both physical correctness and transparency. Code is available at https://github.com/wenjiewang3/PhysicsFromVideo.
Video diffusion models generate visually compelling content but routinely violate elementary physics when the subject involves fluids: liquid columns break apart in mid-air, container water levels fail to rise as liquid is poured in, and splashes disperse without regard to momentum or gravity. We attribute this gap to the fact that large-scale video-text corpora contain almost no explicit motion supervision, so models learn to imitate fluid appearance rather than dynamics. We address this with two contributions. First, we build a physics-simulation fluid dataset combining 1,638 MPM-simulated pouring/sloshing videos with 2,320 keyword-filtered real pouring videos mined from stock footage, plus two held-out test sets: a 1,515-video real-video benchmark and an 18-prompt text-to-first-frame generalization benchmark. Second, we introduce a dual-stream image-to-video architecture built on a pretrained diffusion-transformer video generator. It augments the standard RGB decoder with a lightweight Optical-Flow Decoder branch trained with explicit end-point-error and smoothness losses, fused into the RGB stream via zero-initialized convolutions so the pretrained backbone starts undisturbed. Only the two decoders are updated; the encoder, temporal transformer, and text encoder remain frozen. Across two model scales (1.3B and 14B) and two test sets, our method improves VideoPhy-2 Physical-Commonsense and Video-Quality scores over the frozen backbone by up to 8.75 and 4.65 points, outperforms a leading open competitor, and is preferred by human raters in a blind study. A direct optical-flow read-out evaluation further shows an end-point error as low as 0.54 pixels in-distribution, confirming the model has internalized a coherent motion prior rather than merely improving surface appearance.