Particle image velocimetry (PIV) and optical flow velocimetry (OFV) are important velocity measurement methods in the field of fluid dynamics. Nevertheless, the conventional cross correlation-based PIV method is beset by diminished resolution, while the OFV method exhibits computational sluggishness and susceptibility to noise. These constraints have somewhat delimited the applicability of PIV and OFV techniques. Recent attempts have introduced deep learning-based methods for analyzing PIV images, offering high-resolution velocity fields with computational efficiency, but their accuracy needs improvement. This study proposes four neural networks based on the well-established FlowNetS. They incorporate two distinct velocity constraints, namely, first-order velocity smoothing regularization and second-order grad (curl)–grad (div) regularization. In the networks, these constraints are used either independently or in combination with optical flow conservation (OFC). The performances of the networks have been assessed on six different flow configurations, and the results show that the network with the second-order regularization markedly outperforms the original network across all flows, demonstrating an enhanced capacity to capture larger-scale vortices. The network with the first-order regularization also exhibits superior performance compared to the original network except in the case of cylinder flow. Unexpectedly, the introduction of the OFC constraints results in a decline in network performance. This anomaly may stem from the network's inherent capability to capture optical flow features, rendering the OFC constraint less effective in providing guidance. In summary, this study underscores the substantial potential of neural networks incorporated with judicious physical constraints in PIV applications, enabling the determination of high-resolution, high-accuracy flow fields.

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