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Koopman operators linearize nonlinear dynamical systems, making their spectral information of crucial interest. Numerous algorithms have been developed to approximate these spectral properties, and dynamic mode decomposition (DMD) stands out as the poster child of projection-based methods. Although the Koopman operator itself is linear, the fact that it acts in an infinite-dimensional space of observables poses challenges. These include spurious modes, essential spectra, and the verification of Koopman mode decompositions. While recent work has addressed these challenges for deterministic systems, there remains a notable gap in verified DMD methods for stochastic systems, where the Koopman operator measures the expectation of observables. We show that it is necessary to go beyond expectations to address these issues. By incorporating variance into the Koopman framework, we address these challenges. Through an additional DMD-type matrix, we approximate the sum of a squared residual and a variance term, each of which can be approximated individually using batched snapshot data. This allows verified computation of the spectral properties of stochastic Koopman operators, controlling the projection error. We also introduce the concept of variance-pseudospectra to gauge statistical coherency. Finally, we present a suite of convergence results for the spectral information of stochastic Koopman operators. Our study concludes with practical applications using both simulated and experimental data. In neural recordings from awake mice, we demonstrate how variance-pseudospectra can reveal physiologically significant information unavailable to standard expectation-based dynamical models.

 

Abstract Cyclical variations of the solar magnetic fields, and hence the level of solar activity, are among the top interests of space weather research. Surface flows in global-scale, in particular differential rotation and meridional flows, play important roles in the solar dynamo that describes the origin and variation of solar magnetic fields. In principle, differential rotation is the fundamental cause of dipole field formation and emergence, and meridional flows are the surface component of a longitudinal circulation that brings decayed field from low latitudes to polar regions. Such flows are key inputs and constraints of observational and modeling studies of solar cycles. Here, we present two methods, local correlation tracking (LCT) and machine learning-based self-supervised optical flow methods, to measure differential rotation and meridional flows from full-disk magnetograms that probe the photosphere and $\text{H}\alpha$ H α images that probe the chromosphere, respectively. LCT is robust in deriving photospheric flows using magnetograms. However, we found that it failed to trace flows using time-sequence $\text{H}\alpha $ H α data because of the strong dynamics of traceable features. The optical flow methods handle $\text{H}\alpha $ H α data better to measure the chromospheric flow fields. We found that the differential rotation from photospheric and chromospheric measurements shows a strong correlation with a maximum of $2.85~\upmu \text{rad}\,\text{s}^{-1}$ 2.85 μrad s − 1 at the equator and the accuracy holds until $60^{\circ }$ 60 ∘ for the MDI and $\text{H}\alpha$ H α , $75^{\circ }$ 75 ∘ for the HMI dataset. On the other hand, the meridional flow deduced from the chromospheric measurement shows a similar trend as the concurrent photospheric measurement within $60^{\circ }$ 60 ∘ with a maximum of $20~\text{m}\,\text{s}^{-1}$ 20 m s − 1 at $40^{\circ }$ 40 ∘ in latitude. Furthermore, the measurement uncertainties are discussed. 

Magnetic reconnection is regarded as the mechanism for the rapid release of magnetic energy stored in active regions during solar flares, and quantitative measurements of the magnetic reconnection rate are essential for understanding solar flares. In the context of the standard two-ribbon flare model, we derive the coronal magnetic reconnection rate of the M6.5 flare on 2015 June 22 in two terms, reconnection flux change rate and reconnection electric field, both of which can be obtained from observations of the flare morphology. Data used include a sequence of chromospheric Hαimages with unprecedented resolution during the flare from the Visual Imaging Spectrometer of the Goode Solar Telescope (GST) at the Big Bear Solar Observatory and a preflare line-of-sight photospheric magnetogram from the GST Near-InfraRed Imaging Spectropolarimeter along with hard X-ray data from the Ramaty High Energy Solar Spectroscopic Imager. The temporal correlation between the magnetic reconnection rate and nonthermal emission is found, and the variation of the reconnection electric field is mainly determined by the ribbon speed, not by the local magnetic field encountered by the ribbon front. Spatially, the hard X-ray source overlaps with the location of the strongest electric field obtained at the same time. The ribbon motion shows abundant fine structures, including a local acceleration at the location of a light bridge with a weaker magnetic field.

 

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 for SDNN are 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 that the learning capability of SDNN is better than those of the ML algorithms. Finally, a 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.