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.

翻译：从视频直接推断连续介质模型面临两个难题：记录的场是未经校准的图像强度而非物理状态，且对含噪声帧的直接数值微分不稳定。我们开发了一个视频到偏微分方程的流水线，将墨水羽流的灰度记录转换为归一化标量场 $u(x,y,t)$，通过强度加权质心从内在扩散中分离出整体漂移 $\mathbf{v}(t)$，并利用弱形式稀疏回归识别有效输运定律。条件化、阈值扫描和随机中心诊断表明，过完备库存在强共线性；因此搜索局限于紧凑的梯度基库。系数通过逆向物理信息网络优化，并基于正向推演重新校准，利用时间块自举法量化不确定性。所选简化模型 $u_t+\mathbf v(t)\!\cdot\!\nabla u = 9.005\,|\nabla u|^{2}+0.666\,Δu$ 在保留帧上优于平流-扩散基线，保持正的拉普拉斯系数，并允许通过Cole-Hopf变换简化为线性平流-扩散方程。该框架表明，当发现、校准和不确定性被作为不同阶段处理时，未校准的视觉数据可产生紧凑、可预测且结构可解释的连续介质模型。