Extracting information on fluid motion directly from images is challenging. Fluid flow represents a complex dynamic system governed by the Navier-Stokes equations. General optical flow methods are typically designed for rigid body motion, and thus struggle if applied to fluid motion estimation directly. Further, optical flow methods only focus on two consecutive frames without utilising historical temporal information, while the fluid motion (velocity field) can be considered a continuous trajectory constrained by time-dependent partial differential equations (PDEs). This discrepancy has the potential to induce physically inconsistent estimations. Here we propose an unsupervised learning based prediction-correction scheme for fluid flow estimation. An estimate is first given by a PDE-constrained optical flow predictor, which is then refined by a physical based corrector. The proposed approach outperforms optical flow methods and shows competitive results compared to existing supervised learning based methods on a benchmark dataset. Furthermore, the proposed approach can generalize to complex real-world fluid scenarios where ground truth information is effectively unknowable. Finally, experiments demonstrate that the physical corrector can refine flow estimates by mimicking the operator splitting method commonly utilised in fluid dynamical simulation.
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