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While humans can successfully navigate using abstractions, ignoring details that are irrelevant to the task at hand, most of the existing approaches in robotics require detailed environment representations which consume a significant amount of sensing, computing, and storage; these issues become particularly important in resource-constrained settings with limited power budgets. Deep learning methods can learn from prior experience to abstract knowledge from novel environments, and use it to more efficiently execute tasks such as frontier exploration, object search, or scene understanding. We propose BoxMap, a Detection-Transformer-based architecture that takes advantage of the structure of the sensed partial environment to update a topological graph of the environment as a set of semantic entities (rooms and doors) and their relations (connectivity). The predictions from low-level measurements can be leveraged to achieve high-level goals with lower computational costs than methods based on detailed representations. As an example application, we consider a robot equipped with a 2-D laser scanner tasked with exploring a residential building. Our BoxMap representation scales quadratically with the number of rooms (with a small constant), resulting in significant savings over a full geometric map. Moreover, our high-level topological representation results in 30.9% shorter trajectories in the exploration task with respect to a standard method. Code is available at: bit.ly/3F6w2Yl. 

Deep learning methods have been widely used in robotic applications, making learning-enabled control design for complex nonlinear systems a promising direction. Although deep reinforcement learning methods have demonstrated impressive empirical performance, they lack the stability guarantees that are important in safety-critical situations. One way to provide these guarantees is to learn Lyapunov certificates alongside control policies. There are three related problems: 1) verify that a given Lyapunov function candidate satisfies the conditions for a given controller on a region, 2) find a valid Lyapunov function and controller on a given region, and 3) find a valid Lyapunov function and a controller such that the region of attraction is as large as possible. Previous work has shown that if the dynamics are piecewise linear, it is possible to solve problem 1) and 2) by solving a Mixed-Integer Linear Program (MILP). In this work, we build upon this method by proposing a Lyapunov neural network that considers monotonicity over half spaces in different directions. We 1) propose a specific choice of Lyapunov function architecture that ensures non-negativity and a unique global minimum by construction, and 2) show that this can be leveraged to find the controller and Lyapunov certificates faster and with a larger valid region by maximizing the size of a square inscribed in a given level set. We apply our method to a 2D inverted pendulum, unicycle path following, a 3-D feedback system, and a 4-D cart pole system, and demonstrate it can shorten the training time by half compared to the baseline, as well as find a larger ROA.