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Context. The inverse Evershed flow (IEF) is a mass motion towards sunspots at chromospheric heights. Aims. We combined high-resolution observations of NOAA 12418 from the Dunn Solar Telescope and vector magnetic field measurements from the Helioseismic and Magnetic Imager (HMI) to determine the driver of the IEF. Methods. We derived chromospheric line-of-sight (LOS) velocities from spectra of H α and Ca  II IR. The HMI data were used in a non-force-free magnetic field extrapolation to track closed field lines near the sunspot in the active region. We determined their length and height, located their inner and outer foot points, and derived flow velocities along them. Results. The magnetic field lines related to the IEF reach on average a height of 3 megameter (Mm) over a length of 13 Mm. The inner (outer) foot points are located at 1.2 (1.9) sunspot radii. The average field strength difference Δ B between inner and outer foot points is +400 G. The temperature difference Δ T is anti-correlated with Δ B with an average value of −100 K. The pressure difference Δ p is dominated by Δ B and is primarily positive with a driving force towards the inner foot points of 1.7 kPa on average. The velocities predicted from Δ p reproduce the LOS velocities of 2–10 km s −1 with a square-root dependence. Conclusions. We find that the IEF is driven along magnetic field lines connecting network elements with the outer penumbra by a gas pressure difference that results from a difference in field strength as predicted by the classical siphon flow scenario. 

The paradigm of differentiable programming has significantly enhanced the scope of machine learning via the judicious use of gradient-based optimization. However, standard differentiable programming methods (such as autodiff) typically require that the machine learning models be differentiable, limiting their applicability. Our goal in this paper is to use a new, principled approach to extend gradient-based optimization to functions well modeled by splines, which encompass a large family of piecewise polynomial models. We derive the form of the (weak) Jacobian of such functions and show that it exhibits a block-sparse structure that can be computed implicitly and efficiently. Overall, we show that leveraging this redesigned Jacobian in the form of a differentiable" layer''in predictive models leads to improved performance in diverse applications such as image segmentation, 3D point cloud reconstruction, and finite element analysis. We also open-source the code at\url {https://github. com/idealab-isu/DSA}.