Element learning: a systematic approach of accelerating finite element-type methods via machine learning, with applications to radiative transfer

In this paper, we propose a systematic approach for accelerating finite element-type methods by machine learning for the numerical solution of partial differential equations (PDEs). The main idea is to use a neural network to learn the solution map of the PDEs and to do so in an element-wise fashion. This map takes input of the element geometry and the PDEs' parameters on that element, and gives output of two operators -- (1) the in2out operator for inter-element communication, and (2) the in2sol operator (Green's function) for element-wise solution recovery. A significant advantage of this approach is that, once trained, this network can be used for the numerical solution of the PDE for any domain geometry and any parameter distribution without retraining. Also, the training is significantly simpler since it is done on the element level instead on the entire domain. We call this approach element learning. This method is closely related to hybridizbale discontinuous Galerkin (HDG) methods in the sense that the local solvers of HDG are replaced by machine learning approaches. Numerical tests are presented for an example PDE, the radiative transfer equation, in a variety of scenarios with idealized or realistic cloud fields, with smooth or sharp gradient in the cloud boundary transition. Under a fixed accuracy level of $10^{-3}$ in the relative $L^2$ error, and polynomial degree $p=6$ in each element, we observe an approximately 5 to 10 times speed-up by element learning compared to a classical finite element-type method

Particle-Continuum Multiscale Modeling of Sea Ice Floes

Sea ice profoundly influences the polar environment and the global climate. Traditionally, Sea ice has been modeled as a continuum under Eulerian coordinates to describe its large-scale features, using, for instance, viscous-plastic rheology. Recently, Lagrangian particle models, also known as the discrete element method (DEM) models, have been utilized for characterizing the motion of individual sea ice fragments (called floes) at scales of 10 km and smaller, especially in marginal ice zones. This paper develops a multiscale model that couples the particle and the continuum systems to facilitate an effective representation of the dynamical and statistical features of sea ice across different scales. The multiscale model exploits a Boltzmann-type system that links the particle movement with the continuum equations. For the small-scale dynamics, it describes the motion of each sea ice floe. Then, as the large-scale continuum component, it treats the statistical moments of mass density and linear and angular velocities. The evolution of these statistics affects the motion of individual floes, which in turn provides bulk feedback that adjusts the large-scale dynamics. Notably, the particle model characterizing the sea ice floes is localized and fully parallelized, in a framework that is sometimes called superparameterization, which significantly improves computation efficiency. Numerical examples demonstrate the effective performance of the multiscale model. Additionally, the study demonstrates that the multiscale model has a linear-order approximation to the truth model

Conservation laws for potential vorticity in a salty ocean or cloudy atmosphere

One of the most important conservation laws in atmospheric and oceanic science is conservation of potential vorticity. The original derivation is approximately a century old, in the work of Rossby and Ertel, and it is related to the celebrated circulation theorems of Kelvin and Bjerknes. However, the laws apply to idealized fluids, and extensions to more realistic scenarios have been problematic. Here, these laws are extended to hold with additional fundamental complexities, including salinity in the ocean, or moisture and clouds in the atmosphere. In the absence of these additional complexities, it is known that potential vorticity is conserved following each fluid parcel; here, for a salty ocean or cloudy atmosphere, the general conserved quantity is potential vorticity integrated over certain pancake-shaped volumes. Furthermore, the conservation laws are also related to a symmetry in the Lagrangian, which brings a connection to the symmetry-conservation relationships seen in other areas of physics.Comment: 11 pages, 3 figure

Evaluating behavioral responses of nesting lesser snow geese to unmanned aircraft surveys

Unmanned aircraft systems (UAS) are relatively new technologies gaining popularity among wildlife biologists. As with any new tool in wildlife science, operating protocols must be developed through rigorous protocol testing. Few studies have been conducted that quantify the impacts UAS may have on unhabituated individuals in the wild using standard aerial survey protocols. We evaluated impacts of unmanned surveys by measuring UAS-induced behavioral responses during the nesting phase of lesser snow geese (Anser caerulescens caerulescens) in Wapusk National Park, Manitoba, Canada. We conducted surveys with a fixed-wing Trimble UX5 and monitored behavioral changes via discreet surveillance cameras at 25 nests. Days with UAS surveys resulted in decreased resting and increased nest maintenance, low scanning, high scanning, head-cocking and off-nest behaviors when compared to days without UAS surveys. In the group of birds flown over, head-cocking for overhead vigilance was rarely seen prior to launch or after landing (mean estimates 0.03% and 0.02%, respectively) but increased to 0.56% of the time when the aircraft was flying overhead suggesting that birds were able to detect the aircraft during flight. Neither UAS survey altitude nor launch distance alone in this study was strong predictors of nesting behaviors, although our flight altitudes (≥75 m above ground level) were much higher than previously published behavioral studies. Synthesis and applications: The diversity of UAS models makes generalizations on behavioral impacts difficult, and we caution that researchers should design UAS studies with knowledge that some minimal disturbance is likely to occur. We recommend flight designs take potential behavioral impacts into account by increasing survey altitude where data quality requirements permit. Such flight designs should consider a priori knowledge of focal species’ behavioral characteristics. Research is needed to determine whether any such disturbance is a result of visual or auditory stimuli

Convergence of Rain Process Models to Point Processes

A moisture process with dynamics that switch after hitting a threshold gives rise to a rainfall process. This rainfall process is characterized by its random holding times for dry and wet periods. On average, the holding times for the wet periods are much shorter than the dry. Here convergence is shown for the rain fall process to a point process that is a spike train. The underlying moisture process for the point process is a threshold model with a teleporting boundary condition. This approximation allows simplification of the model with many exact formulas for statistics. The convergence is shown by a Fokker-Planck derivation, convergence in mean-square with respect to continuous functions, of the moisture process, and convergence in mean-square with respect to generalized functions, of the rain process.Comment: 11 pages, 2 figure