ABSTRACT Obtaining high-quality magnetic and velocity fields through Stokes inversion is crucial in solar physics. In this paper, we present a new deep learning method, named Stacked Deep Neural Networks (SDNN), for inferring line-of-sight (LOS) velocities and Doppler widths from Stokes profiles collected by the Near InfraRed Imaging Spectropolarimeter (NIRIS) on the 1.6 m Goode Solar Telescope (GST) at the Big Bear Solar Observatory (BBSO). The training data of SDNN is prepared by a Milne-Eddington (ME) inversion code used by BBSO. We quantitatively assess SDNN, comparing its inversion results with those obtained by the ME inversion code and related machine learning (ML) algorithms such as multiple support vector regression, multilayer perceptrons and a pixel-level convolutional neural network. Major findings from our experimental study are summarized as follows. First, the SDNN-inferred LOS velocities are highly correlated to the ME-calculated ones with the Pearson product-moment correlation coefficient being close to 0.9 on average. Second, SDNN is faster, while producing smoother and cleaner LOS velocity and Doppler width maps, than the ME inversion code. Third, the maps produced by SDNN are closer to ME's maps than those from the related ML algorithms, demonstrating the better learning capability of SDNN than the ML algorithms. Finally, comparison between the inversion results of ME and SDNN based on GST/NIRIS and those from the Helioseismic and Magnetic Imager on board the Solar Dynamics Observatory in flare-prolific active region NOAA 12673 is presented. We also discuss extensions of SDNN for inferring vector magnetic fields with empirical evaluation. INTRODUCTION Obtaining high-quality magnetic and velocity fields through Stokes inversion is crucial in solar physics (Nejezchleba 1998; Maurya & Ambastha 2010; Bobra & Couvidat 2015).

More competent learning models are demanded for data processing due to increasingly greater amounts of data available in applications. Data that we encounter often have certain embedded sparsity structures. That is, if they are represented in an appropriate basis, their energies can concentrate on a small number of basis functions. This paper is devoted to a numerical study of adaptive approximation of solutions of nonlinear partial differential equations whose solutions may have singularities, by deep neural networks (DNNs) with a sparse regularization with multiple parameters. Noting that DNNs have an intrinsic multi-scale structure which is favorable for adaptive representation of functions, by employing a penalty with multiple parameters, we develop DNNs with a multi-scale sparse regularization (SDNN) for effectively representing functions having certain singularities. We then apply the proposed SDNN to numerical solutions of the Burgers equation and the Schr\"odinger equation. Numerical examples confirm that solutions generated by the proposed SDNN are sparse and accurate.