  We propose a method using supervised machine learning to estimate velocity
fields from particle images having missing regions due to experimental
limitations. As a first example, a velocity field around a square cylinder at
Reynolds number of ${\rm Re}_D=300$ is considered. To train machine learning
models, we utilize artificial particle images (APIs) as the input data, which
mimic the images of the particle image velocimetry (PIV). The output data are
the velocity fields, and the correct answers for them are given by a direct
numerical simulation (DNS). We examine two types of the input data: APIs
without missing regions (i.e., full APIs) and APIs with missing regions (lacked
APIs). The missing regions in the lacked APIs are assumed following the exact
experimental situation in our wind tunnel setup. The velocity fields estimated
from both full and lacked APIs are in great agreement with the reference DNS
data in terms of various statistical assessments. We further apply these
machine learned models trained with the DNS data to experimental particle
images so that their applicability to the exact experimental situation can be
investigated. The velocity fields estimated by the machine learned models
contain approximately 40 folds denser data than that with the conventional
cross-correlation method. This finding suggests that we may be able to obtain
finer and hidden structures of the flow field which cannot be resolved with the
conventional cross-correlation method. We also found that even the complex flow
structures are hidden due to the alignment of two square cylinders, the machine
learned model is able to estimate the field in the missing region reasonably
well. The present results indicate a great potential of the proposed machine
learning based method as a new data reconstruction method for PIV.
