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This paper develops a treetopological local mesh refinement (TLMR) method on Cartesian grids for the simulation of bioinspired flow with multiple moving objects. The TLMR nests refinement mesh blocks of structured grids to the target regions and arrange the blocks in a tree topology. The method solves the timedependent incompressible flow using a fractionalstep method and discretizes the NavierStokes equation using a finitedifference formulation with an immersed boundary method to resolve the complex boundaries. When iteratively solving the discretized equations across the coarse and fine TLMR blocks, for better accuracy and faster convergence, the momentum equation is solved on all blocks simultaneously, while the Poisson equation is solved recursively from the coarsest block to the finest ones. When the refined blocks of the same block are connected, the parallel Schwarz method is used to iteratively solve both the momentum and Poisson equations. Convergence studies show that the algorithm is secondorder accurate in space for both velocity and pressure, and the developed mesh refinement technique is benchmarked and demonstrated by several canonical flow problems. The TLMR enables a fast solution to an incompressible flow problem with complex boundaries or multiple moving objects. Various bioinspired flows of multiple moving objects show that the solver can save over 80% computational time, proportional to the grid reduction when refinement is applied.more » « less

Equivariant representation is necessary for the brain and artificial perceptual systems to faithfully represent the stimulus under some (Lie) group transformations. However, it remains unknown how recurrent neural circuits in the brain represent the stimulus equivariantly, nor the neural representation of abstract group operators. The present study uses a onedimensional (1D) translation group as an example to explore the general recurrent neural circuit mechanism of the equivariant stimulus representation. We found that a continuous attractor network (CAN), a canonical neural circuit model, selfconsistently generates a continuous family of stationary population responses (attractors) that represents the stimulus equivariantly. Inspired by the Drosophila’s compass circuit, we found that the 1D translation operators can be represented by extra speed neurons besides the CAN, where speed neurons’ responses represent the moving speed (1D translation group parameter), and their feedback connections to the CAN represent the translation generator (Lie algebra). We demonstrated that the network responses are consistent with experimental data. Our model for the first time demonstrates how recurrent neural circuitry in the brain achieves equivariant stimulus representation.more » « less

The activity of the grid cell population in the medial entorhinal cortex (MEC) of the mammalian brain forms a vector representation of the selfposition of the animal. Recurrent neural networks have been proposed to explain the properties of the grid cells by updating the neural activity vector based on the velocity input of the animal. In doing so, the grid cell system effectively performs path integration. In this paper, we investigate the algebraic, geometric, and topological properties of grid cells using recurrent network models. Algebraically, we study the Lie group and Lie algebra of the recurrent transformation as a representation of selfmotion. Geometrically, we study the conformal isometry of the Lie group representation where the local displacement of the activity vector in the neural space is proportional to the local displacement of the agent in the 2D physical space. Topologically, the compact abelian Lie group representation automatically leads to the torus topology commonly assumed and observed in neuroscience. We then focus on a simple nonlinear recurrent model that underlies the continuous attractor neural networks of grid cells. Our numerical experiments show that conformal isometry leads to hexagon periodic patterns in the grid cell responses and our model is capable of accurate path integration.more » « less

In this work, numerical simulations are employed to study hydrodynamic interactions in troutlike threedimensional(3D) fish bodies arranged in vertical and horizontal planes. The fish body is modeled on a juvenile rainbow trout (Oncorhynchus mykiss) and is imposed on a traveling wave to mimic trout swimming. Three typical minimal schools are studied, including the inline, the sidebyside, and the vertical school. A sharp interface immersedboundarybased incompressible NavierStrokes flow solver is then used to quantitively simulate the resulting flow and hydrodynamic performance of the schools. The results show that the hydrodynamic efficiency of the leading fish in the inline school increases by 5.28%, and the thrust production and efficiency of the sidebyside school are enhanced by 2.28% and 3.86%, respectively. Besides, the thrust production of the vertical school increases by 21.6%. The results suggest great potential in exploiting the hydrodynamic benefits in fish schools arranged in threedimensional space.more » « less

Despite major improvements in weather and climate modelling and substantial increases in remotely sensed observations, drought prediction remains a major challenge. After a review of the existing methods, we discuss major research gaps and opportunities to improve drought prediction. We argue that current approaches are topdown, assuming that the process(es) and/or driver(s) are known—i.e. starting with a model and then imposing it on the observed events (reality). With the help of an experiment, we show that there are opportunities to develop bottomup drought prediction models—i.e. starting from the reality (here, observed events) and searching for model(s) and driver(s) that work. Recent advances in artificial intelligence and machine learning provide significant opportunities for developing bottomup drought forecasting models. Regardless of the type of drought forecasting model (e.g. machine learning, dynamical simulations, analogue based), we need to shift our attention to robustness of theories and outputs rather than eventbased verification. A shift in our focus towards quantifying the stability of uncertainty in drought prediction models, rather than the goodness of fit or reproducing the past, could be the first step towards this goal. Finally, we highlight the advantages of hybrid dynamical and statistical models for improving current drought prediction models. This article is part of the Royal Society Science+ meeting issue ‘Drought risk in the Anthropocene’.more » « less

Abstract We studied the magnetic excitations in the quasionedimensional (q1D) ladder subsystem of Sr 14−x Ca x Cu 24 O 41 (SCCO) using Cu L 3 edge resonant inelastic Xray scattering (RIXS). By comparing momentumresolved RIXS spectra with high ( x = 12.2) and without ( x = 0) Ca content, we track the evolution of the magnetic excitations from collective twotriplon (2 T) excitations ( x = 0) to weaklydispersive gapped modes at an energy of 280 meV ( x = 12.2). Density matrix renormalization group (DMRG) calculations of the RIXS response in the doped ladders suggest that the flat magnetic dispersion and damped excitation profile observed at x = 12.2 originates from enhanced hole localization. This interpretation is supported by polarizationdependent RIXS measurements, where we disentangle the spinconserving Δ S = 0 scattering from the predominant Δ S = 1 spinflip signal in the RIXS spectra. The results show that the lowenergy weight in the Δ S = 0 channel is depleted when Sr is replaced by Ca, consistent with a reduced carrier mobility. Our results demonstrate that offladder impurities can affect both the lowenergy magnetic excitations and superconducting correlations in the CuO 4 plaquettes. Finally, our study characterizes the magnetic and charge fluctuations in the phase from which superconductivity emerges in SCCO at elevated pressures.more » « less

Algorithmic decisions made by machine learning models in highstakes domains may have lasting impacts over time. However, naive applications of standard fairness criterion in static settings over temporal domains may lead to delayed and adverse effects. To understand the dynamics of performance disparity, we study a fairness problem in Markov decision processes (MDPs). Specifically, we propose return parity, a fairness notion that requires MDPs from different demographic groups that share the same state and action spaces to achieve approximately the same expected timediscounted rewards. We first provide a decomposition theorem for return disparity, which decomposes the return disparity of any two MDPs sharing the same state and action spaces into the distance between groupwise reward functions, the discrepancy of group policies, and the discrepancy between state visitation distributions induced by the group policies. Motivated by our decomposition theorem, we propose algorithms to mitigate return disparity via learning a shared group policy with state visitation distributional alignment using integral probability metrics. We conduct experiments to corroborate our results, showing that the proposed algorithm can successfully close the disparity gap while maintaining the performance of policies on two realworld recommender system benchmark datasets.more » « less