The numerical solution of partial differential equations (PDEs) is challenging because of the need to resolve spatiotemporal features over wide length- and timescales. Often, it is computationally intractable to resolve the finest features in the solution. The only recourse is to use approximate coarse-grained representations, which aim to accurately represent long-wavelength dynamics while properly accounting for unresolved small-scale physics. Deriving such coarse-grained equations is notoriously difficult and often ad hoc. Here we introduce data-driven discretization, a method for learning optimized approximations to PDEs based on actual solutions to the known underlying equations. Our approach uses neural networks to estimate spatial derivatives, which are optimized end to end to best satisfy the equations on a low-resolution grid. The resulting numerical methods are remarkably accurate, allowing us to integrate in time a collection of nonlinear equations in 1 spatial dimension at resolutions 4× to 8× coarser than is possible with standard finite-difference methods.

A key challenge in metasurface design is the development of algorithms that can effectively and efficiently produce high performance devices. Design methods based on iterative optimization can push the performance limits of metasurfaces, but they require extensive computational resources that limit their implementation to small numbers of microscale devices. We show that generative neural networks can train from images of periodic, topology-optimized metagratings to produce high-efficiency, topologically complex devices operating over a broad range of deflection angles and wavelengths. Further iterative optimization of these designs yields devices with enhanced robustness and efficiencies, and these devices can be utilized as additional training data for network refinement. In this manner, generative networks can be trained, with a onetime computation cost, and used as a design tool to facilitate the production of near-optimal, topologically-complex device designs. We envision that such data-driven design methodologies can apply to other physical sciences domains that require the design of functional elements operating across a wide parameter space.

High-contrast coronagraphy will be needed to image and characterize faint extra-solar planetary systems. Coronagraphy is a rapidly evolving field, and many enhanced alternatives to the classical Lyot coronagraph have been proposed in the past ten years. Here, we discuss the operation of the vector vortex coronagraph, which is one of the most efficient possible coronagraphs. We first present recent laboratory results, and then first light observations at the Palomar observatory. Our nearinfrared H-band (centered at ∼ 1.65µm) and K-band (centered at ∼ 2.2µm) vector vortex devices demonstrated excellent contrast results in the lab, down to ∼ 10 −6 at an angular separation of ∼ 3λ/d. On sky, we detected a brown dwarf companion 3000 times fainter than its host star (HR 7672) in the K s band (centered at ∼ 2.15µm), at an angular separation of ∼ 2.5λ/d. Current and next-generation high-contrast instruments can directly benefit from the demonstrated capabilities of such a vector vortex: simplicity, small inner working angle, high optical throughput (> 90%), and maximal off-axis discovery space.

low-earth orbit view only a small fraction of the earth's surface at once so that many observations must be combined to obtain global-scale information. It is also a major technical and logistical challenge to keep calibrated photometers continuously in orbit on decadal timescales. The earthshine, or "ashen light", is the glow of the "dark" part of the lunar disk visible to a nighttime observer. It is sunlight reflected from the earth and retroflected from the lunar surface, and so offers an alternative route to studying the earth's reflectance. Earthshine data are complementary to existing satellite data in that the coverage is instantaneous and hemispheric in scale. Because the earth's phase as seen from the moon is supplementary to that of the moon seen from the earth (i.e., the earth is nearly "full" when the moon is a thin crescent), the instantaneous intensity of the earthshine near the new moon samples almost half of the earth. For over a quarter century, beginning in 1926, Danjon and his followers [Danjon, 1928[Danjon, , 1954Dubois, 1947] performed regular earthshine observations from France. We have reinvigorated and modernized this nearly forgotten way of measuring the earth's albedo.Our earthshine coronagraph is a 6-inch refractive telescope with a cooled AP-7 512x512 pixel CCD camera; our fiducial regions comprise about 100 pixels each. We mask the bright portion of the lunar disk (i.e., "moonshine") with a neutral density filter of transmission • 10 -5, and correct the measured earthshine intensity for the effects of moonshine scattered in the atmosphere and optical train.