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This paper investigates the problem of informative path planning for a mobile robotic sensor network in spatially temporally distributed mapping. The robots are able to gather noisy measurements from an area of interest during their movements to build a Gaussian process (GP) model of a spatio-temporal field. The model is then utilized to predict the spatio-temporal phenomenon at different points of interest. To spatially and temporally navigate the group of robots so that they can optimally acquire maximal information gains while their connectivity is preserved, we propose a novel multi-step prediction informative path planning optimization strategy employing our newly defined local cost functions. By using the dual decomposition method, it is feasible and practical to effectively solve the optimization problem in a distributed manner. The proposed method was validated through synthetic experiments utilizing real-world data sets. 

The deep operator network (DeepONet) architecture is a promising approach for learning functional operators, that can represent dynamical systems described by ordinary or partial differential equations. However, it has two major limitations, namely its failures to account for initial conditions and to guarantee the temporal causality – a fundamental property of dynamical systems. This paper proposes a novel causal deep operator network (Causal-DeepONet) architecture for incorporating both the initial condition and the temporal causality into data-driven learning of dynamical systems, overcoming the limitations of the original DeepONet approach. This is achieved by adding an independent root network for the initial condition and independent branch networks conditioned, or switched on/off, by time-shifted step functions or sigmoid functions for expressing the temporal causality. The proposed architecture was evaluated and compared with two baseline deep neural network methods and the original DeepONet method on learning the thermal dynamics of a room in a building using real data. It was shown to not only achieve the best overall prediction accuracy but also enhance substantially the accuracy consistency in multistep predictions, which is crucial for predictive control. 

Abstract This paper develops new techniques for studying smooth dynamical systems in the presence of a Carnot–Carathéodory metric. Principally, we employ the theory of Margulis and Mostow, Métivier, Mitchell, and Pansu on tangent cones to establish resonances between Lyapunov exponents. We apply these results in three different settings. First, we explore rigidity properties of smooth dominated splittings for Anosov diffeomorphisms and flows via associated smooth Carnot–Carathéodory metrics. Second, we obtain local rigidity properties of higher hyperbolic rank metrics in a neighborhood of a locally symmetric one. For the latter application we also prove structural stability of the Brin–Pesin asymptotic holonomy group for frame flows. Finally, we obtain local rigidity properties for uniform lattice actions on the ideal boundary of quaternionic and octonionic symmetric spaces. 

A Riemannian manifold $$M$$ has higher hyperbolic rank if every geodesic has a perpendicular Jacobi field making sectional curvature $-1$ with the geodesic. If, in addition, the sectional curvatures of $$M$$ lie in the interval $$[-1,-\frac{1}{4}]$$ and $$M$$ is closed, we show that $$M$$ is a locally symmetric space of rank one. This partially extends work by Constantine using completely different methods. It is also a partial counterpart to Hamenstädt’s hyperbolic rank rigidity result for sectional curvatures $$\leq -1$$ , and complements well-known results on Euclidean and spherical rank rigidity.