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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 the Reynolds number of ReD=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 fold 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 find 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.

We investigate the capability of neural network-based model order reduction, i.e., autoencoder (AE), for fluid flows. As an example model, an AE which comprises of convolutional neural networks and multi-layer perceptrons is considered in this study. The AE model is assessed with four canonical fluid flows, namely: (1) two-dimensional cylinder wake, (2) its transient process, (3) NOAA sea surface temperature, and (4) a cross-sectional field of turbulent channel flow, in terms of a number of latent modes, the choice of nonlinear activation functions, and the number of weights contained in the AE model. We find that the AE models are sensitive to the choice of the aforementioned parameters depending on the target flows. Finally, we foresee the extensional applications and perspectives of machine learning based order reduction for numerical and experimental studies in the fluid dynamics community.

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The recent development of high-performance computing enables us to generate spatiotemporal high-resolution data of nonlinear dynamical systems and to analyze them for deeper understanding of their complex nature. This trend can be found in a wide range of science and engineering communities, which suggests that detailed investigations on efficient data handling in physical science must be required in future. To this end, we introduce the use of convolutional neural networks (CNNs) to achieve an efficient data storage and estimation of scientific big data derived from nonlinear dynamical systems.The CNN is utilized to reconstruct three-dimensional data from a few numbers of two-dimensional sections in a computationally friendly manner. The present model is a combination of two-and three-dimensional CNNs, which allows users to save only some of the two-dimensional sections to reconstruct the volumetric data. As an example of threedimensional data, we consider a fluid flow around a square cylinder at the diameter-based Reynolds number Re D of 300, and show that volumetric fluid flow data can successfully be reconstructed with the present method from as few as five sections. Furthermore, we also propose a combination of the present CNN-based reconstruction with an adaptive sampling-based super-resolution analysis to augment the data compression capability of the present methods. Our report can be a significant bridge toward practical data handling for not only the fluid mechanics field but also a vast range of physical sciences.