Diffractive neural networks leverage the high-dimensional characteristics of electromagnetic (EM) fields for high-throughput computing. However, the existing architectures face challenges in integrating large-scale multidimensional metasurfaces with precise network training and haven't utilized multidimensional EM field coding scheme for super-resolution sensing. Here, we propose diffractive meta-neural networks (DMNNs) for accurate EM field modulation through metasurfaces, which enable multidimensional multiplexing and coding for multi-task learning and high-throughput super-resolution direction of arrival estimation. DMNN integrates pre-trained mini-metanets to characterize the amplitude and phase responses of meta-atoms across different polarizations and frequencies, with structure parameters inversely designed using the gradient-based meta-training. For wide-field super-resolution angle estimation, the system simultaneously resolves azimuthal and elevational angles through x and y-polarization channels, while the interleaving of frequency-multiplexed angular intervals generates spectral-encoded optical super-oscillations to achieve full-angle high-resolution estimation. Post-processing lightweight electronic neural networks further enhance the performance. Experimental results validate that a three-layer DMNN operating at 27 GHz, 29 GHz, and 31 GHz achieves $\sim7\times$ Rayleigh diffraction-limited angular resolution (0.5$^\circ$), a mean absolute error of 0.048$^\circ$ for two incoherent targets within a $\pm 11.5^\circ$ field of view, and an angular estimation throughput an order of magnitude higher (1917) than that of existing methods. The proposed architecture advances high-dimensional photonic computing systems by utilizing inherent high-parallelism and all-optical coding methods for ultra-high-resolution, high-throughput applications.

A Differential Drive Robot (DDR) located inside a circular detection region in the plane wants to escape from it in minimum time. Various robotics applications can be modeled like the previous problem, such as a DDR escaping as soon as possible from a forbidden/dangerous region in the plane or running out from the sensor footprint of an unmanned vehicle flying at a constant altitude. In this paper, we find the motion strategies to accomplish its goal under two scenarios. In one, the detection region moves slower than the DDR and seeks to prevent escape; in another, its position is fixed. We formulate the problem as a zero-sum pursuit-evasion game, and using differential games theory, we compute the players' time-optimal motion strategies. Given the DDR's speed advantage, it can always escape by translating away from the center of the detection region at maximum speed. In this work, we show that the previous strategy could be optimal in some cases; however, other motion strategies emerge based on the player's speed ratio and the players' initial configurations.

A fundamental task in mobile robotics is to keep an agent under surveillance using an autonomous robotic platform equipped with a sensing device. Using differential game theory, we study a particular setup of the previous problem. A Differential Drive Robot (DDR) equipped with a bounded range sensor wants to keep surveillance of an Omnidirectional Agent (OA). The goal of the DDR is to maintain the OA inside its detection region for as much time as possible, while the OA, having the opposite goal, wants to leave the regions as soon as possible. We formulate the problem as a zero-sum differential game, and we compute the time-optimal motion strategies of the players to achieve their goals. We focus on the case where the OA is faster than the DDR. Given the OA's speed advantage, a winning strategy for the OA is always moving radially outwards to the DDR's position. However, this work shows that even though the previous strategy could be optimal in some cases, more complex motion strategies emerge based on the players' speed ratio. In particular, we exhibit that four classes of singular surfaces may appear in this game: Dispersal, Transition, Universal, and Focal surfaces. Each one of those surfaces implies a particular motion strategy for the players.

A fundamental task in mobile robotics is keeping an intelligent agent under surveillance with an autonomous robot as it travels in the environment. This work studies a version of that problem involving one of the most popular vehicle platforms in robotics. In particular, we consider two identical Dubins cars moving on a plane without obstacles. One of them plays as the pursuer, and it is equipped with a limited field-of-view detection region modeled as a semi-infinite cone with its apex at the pursuer's position. The pursuer aims to maintain the other Dubins car, which plays as the evader, as much time as possible inside its detection region. On the contrary, the evader wants to escape as soon as possible. In this work, employing differential game theory, we find the time-optimal motion strategies near the game's end. The analysis of those trajectories reveals the existence of at least two singular surfaces: a Transition Surface and an Evader's Universal Surface. We also found that the barrier's standard construction produces a surface that partially lies outside the playing space and fails to define a closed region, implying that an additional procedure is required to determine all configurations where the evader escapes.

We consider the problem of finding the matching map between two sets of $d$ dimensional vectors from noisy observations, where the second set contains outliers. The matching map is then an injection, which can be consistently estimated only if the vectors of the second set are well separated. The main result shows that, in the high-dimensional setting, a detection region of unknown injection can be characterized by the sets of vectors for which the inlier-inlier distance is of order at least $d^{1/4}$ and the inlier-outlier distance is of order at least $d^{1/2}$. These rates are achieved using the estimated matching minimizing the sum of logarithms of distances between matched pairs of points. We also prove lower bounds establishing optimality of these rates. Finally, we report results of numerical experiments on both synthetic and real world data that illustrate our theoretical results and provide further insight into the properties of the estimators studied in this work.