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Multi-Scale Deep Learning for Estimating Horizontal Velocity Fields on the Solar Surface

The dynamics in the photosphere is governed by the multi-scale turbulent convection termed as granulation and supergranulation. It is important to derive 3-dimensional velocity vectors to understand the nature of the turbulent convection. However, it is difficult to obtain the velocity component perpendicular to the line-of-sight, which corresponds to the horizontal velocity in disk center observations. We developed a convolutional neural network model with a multi-scale deep learning architecture. The method consists of multiple convolutional kernels with various sizes of the receptive fields, and it performs convolution for spatial and temporal axes. The network is trained with data from three different numerical simulations of turbulent convection, and we introduced a coherence spectrum to assess the horizontal velocity fields that were derived at each spatial scale. The multi-scale deep learning method successfully predicts the horizontal velocities for each convection simulation in terms of the global-correlation-coefficient, which is often used for evaluating the prediction accuracy of the methods. The coherence spectrum reveals the strong dependence of the correlation coefficients on the spatial scales. Although coherence spectra are higher than 0.9 for large-scale structures, they drastically decrease to less than 0.3 for small-scale structures wherein the global-correlation-coefficient indicates a high value of approximately 0.95. We determined that this decrease in the coherence spectrum occurs around the energy injection scales. The accuracy for the small-scale structures is not guaranteed solely by the global-correlation-coefficient. To improve the accuracy in small-scales, it is important to improve the loss function for enhancing the small-scale structures and to utilize other physical quantities related to the non-linear cascade of convective eddies as input data.