Coarse-to-fine variational inference with physics-informed deep learning for complex fluid motion estimation

Particle image velocimetry (PIV) is a cornerstone of experimental fluid mechanics, yet existing methods present a trade-off between the robustness of cross-correlation and the resolution of variational optical flow. Although modern deep learning models have bridged this gap, they often rely on computationally expensive, architecturally complex black-box modules that limit efficiency and interpretability. In this work, we introduce a different paradigm: a physics-informed variational inference framework that achieves state-of-the-art performance through a transparent and principled design. Our model replaces the conventional learned cost volume with direct photometric error signals and integrates physical constraints, such as brightness constancy and incompressibility, at every level of a multi-scale pyramid hierarchy. The entire refinement process is guided by a robust, computationally efficient coarse velocity prior, which stabilizes training and enhances final accuracy. On synthetic benchmarks, our method establishes new state-of-the-art accuracy on the most challenging large-displacement flows, including turbulence and surface quasi-geostrophic flow, with improvements of 18% and 40%, respectively, over leading models, while being over 6 times faster. Furthermore, the model demonstrates superior robustness to noise, particle density variations, and occlusions. When applied to real-world riverine data without retraining, it shows outstanding generalization, producing velocity fields that align closely with in situ measurements. This work demonstrates that by grounding deep learning in classical variational principles and explicit physical laws, it is possible to create PIV models that are simultaneously more accurate, efficient, and interpretable.