Previous research on this problem has mainly focused on improving DGMs by either introducing new objective functions or designing more expressive model architectures. With the rapidly growing model complexity and data volume, training deep generative models (DGMs) for better performance has becoming an increasingly more important challenge. We provide rigorous theoretical development, computational methodologies, numerical examples, and MATLAB code for both benchmarks. We prove that the error metric obeys a central limit theorem, develop a streamlined method for performing computations, and place the standard statistical tests used here on a firm theoretical footing. We demonstrate the utility of this benchmark in assessing the performance of stochastic control algorithms. The second benchmark compares the observed distribution of error metric values to the probability density function of the error metric when robot positions are randomly sampled from the target distribution. We also show that the error metric extrema can be used to help choose the swarm size and effective radius of each robot required to achieve a desired level of coverage. The first uses the realizable extrema of the error metric to compute the relative error of an observed swarm distribution. We analyze the theoretical and computational properties of the error metric and propose two benchmarks to which error metric values can be compared. The proposed error metric is continuously sensitive to changes in the swarm distribution, unlike commonly used discretization methods. This work fills that gap by introducing an error metric that provides a quantitative measure of coverage for use with any control scheme. In the field of swarm robotics, the design and implementation of spatial density control laws has received much attention, with less emphasis being placed on performance evaluation. The physical parameter in this work is either the small diffusivity of chemo-attractant or the reciprocal of the flow amplitude in the advection-dominated regime. We present numerical results of DP framework for successful learning and generation of KS dynamics in the presence of laminar and chaotic flows. To reduce computational cost, we develop an iterative divide-and-conquer algorithm to find the optimal transition matrix in the Wasserstein distance. In the training stage, we update the network weights by minimizing a discrete 2-Wasserstein distance between the input and target empirical measures. We utilize the expressiveness of deep neural networks (DNNs) to represent the transform of samples from a given initial (source) distribution to a target distribution at finite time T prior to blowup without assuming invertibility of the transforms. The KS solutions are approximated as empirical measures of particles which self-adapt to the high gradient part of solutions. We study a regularized interacting particle method for computing aggregation patterns and near singular solutions of a Keller-Segal (KS) chemotaxis system in two and three space dimensions, then further develop DeepParticle (DP) method to learn and generate solutions under variations of physical parameters. We close by applying such a method - the blob method for diffusion - to showcase key properties of the dynamics of aggregation-diffusion equations and related singular limits. We also review the range of numerical methods available for simulating solutions, with special attention devoted to recent advances in deterministic particle methods. In the present work, we review known analytical results for aggregation-diffusion equations and consider singular limits of these equations, including the slow diffusion limit, which leads to the constrained aggregation equation, as well as localized aggregation and vanishing diffusion limits, which lead to metastability behavior. These competing forces lead to rich dynamics, including symmetrization, stabilization, and metastability, as well as sharp dichotomies separating well-posedness from finite time blowup. Over the past fifteen years, aggregation-diffusion equations have become widespread in biological applications and have also attracted significant mathematical interest, due to their competing forces at different length scales. Given a large ensemble of interacting particles, driven by nonlocal interactions and localized repulsion, the mean-field limit leads to a class of nonlocal, nonlinear partial differential equations known as aggregation-diffusion equations.
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