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The power of DNN has been successfully demonstrated on a wide variety of high-dimensional problems that cannot be solved by conventional control design methods. These successes also uncover some fundamental and pressing challenges in understanding the representability of deep neural networks for complex and high dimensional input–output relations. Towards the goal of understanding these fundamental questions, we applied an algebraic framework developed in our previous work to analyze ReLU neural network approximation of compositional functions. We prove that for Lipschitz continuous functions, ReLU neural networks have an approximation error upper bound that is a polynomial of the network’s complexity and the compositional features. If the compositional features do not increase exponentially with dimension, which is the case in many applications, the complexity of DNN has a polynomial growth. In addition to function approximations, we also establish ReLU network approximation results for the trajectories of control systems, and for a Lyapunov function that characterizes the domain of attraction. 

Abstract The origin of high-energy galactic cosmic rays is yet to be understood, but some galactic cosmic-ray accelerators can accelerate cosmic rays up to PeV energies. The high-energy cosmic rays are expected to interact with the surrounding material or radiation, resulting in the production of gamma-rays and neutrinos. To optimize for the detection of such associated production of gamma-rays and neutrinos for a given source morphology and spectrum, a multimessenger analysis that combines gamma-rays and neutrinos is required. In this study, we use the Multi-Mission Maximum Likelihood framework with IceCube Maximum Likelihood Analysis software and HAWC Accelerated Likelihood to search for a correlation between 22 known gamma-ray sources from the third HAWC gamma-ray catalog and 14 yr of IceCube track-like data. No significant neutrino emission from the direction of the HAWC sources was found. We report the best-fit gamma-ray model and 90% CL neutrino flux limit from the 22 sources. From the neutrino flux limit, we conclude that, for five of the sources, the gamma-ray emission observed by HAWC cannot be produced purely from hadronic interactions. We report the limit for the fraction of gamma-rays produced by hadronic interactions for these five sources. 

Abstract Name that Neutrinois a citizen science project where volunteers aid in classification of events for the IceCube Neutrino Observatory, an immense particle detector at the geographic South Pole. From March 2023 to September 2023, volunteers did classifications of videos produced from simulated data of both neutrino signal and background interactions.Name that Neutrinoobtained more than 128,000 classifications by over 1800 registered volunteers that were compared to results obtained by a deep neural network machine-learning algorithm. Possible improvements for bothName that Neutrinoand the deep neural network are discussed.