ReachLipBnB: A branch-and-bound method for reachability analysis of neural autonomous systems using Lipschitz bounds

We propose a novel Branch-and-Bound method for reachability analysis of neural networks in both open-loop and closed-loop settings. Our idea is to first compute accurate bounds on the Lipschitz constant of the neural network in certain directions of interest offline using a convex program. We then use these bounds to obtain an instantaneous but conservative polyhedral approximation of the reachable set using Lipschitz continuity arguments. To reduce conservatism, we incorporate our bounding algorithm within a branching strategy to decrease the over-approximation error within an arbitrary accuracy. We then extend our method to reachability analysis of control systems with neural network controllers. Finally, to capture the shape of the reachable sets as accurately as possible, we use sample trajectories to inform the directions of the reachable set over-approximations using Principal Component Analysis (PCA). We evaluate the performance of the proposed method in several open-loop and closed-loop settings

The effects of friction stir welding on microstructure and formability of 7075-T6 sheet

The present research aims to investigate the effects of friction stir welding (FSW) on the microstructure and mechanical properties of 7075-T6 aluminum alloy thin sheets. The welding has been performed perpendicular to the rolling direction of the sheets at three rotational speeds of 600, 1000, and 1600 rpm. From mechanical and microstructural points of view, a suitable welding condition has been recognized at 1000 rpm and welding speed of 50 mm/min. It is observed that with increasing the welding rotational speed more precipitates of the base metal are being dissolved in the metallic matrix and their distribution is more uniform. This phenomenon is recognized as the main mechanism of mechanical properties decline at 1600 rpm. Surprisingly 15,000 h after the welding process, the natural aging could not recover the precipitations within the welding zone of the specimen welded by 1600 rpm rotational speed. The effects of FSW on the three-dimensional forming behavior of the joints have been examined by the hemispherical punch stretching (HPS) method. Forming limit diagrams (FLD) of the weldments and the starting material have been measured and compared to each other. The formability examinations have demonstrated a 40% decline in the forming limit of the welded specimens in comparison to the base metal. The main reasons for the smaller formability of the welded specimens are comprehensively discussed