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The robustness of neural networks is crucial in safety-critical applications, where identifying a reliable input space is essential for effective model selection, robustness evaluation, and the development of reliable control strategies. Most existing robustness verification methods assess the worst-case output under the assumption that the input space is known. However, precisely identifying a verifiable input space , where no adversarial examples exist, is challenging due to the possible high dimensionality, discontinuity, and non-convex nature of the input space. To address this challenge, we propose a novel framework, LEVIS, comprising LEVIS- and LEVIS-. LEVIS- identifies a single, large verifiable ball that intersects at least two boundaries of a bounded region , while LEVIS- systematically captures the entirety of the verifiable space by integrating multiple verifiable balls. Our contributions are fourfold: we introduce a verification framework, LEVIS, incorporating two optimization techniques for computing nearest and directional adversarial points based on mixed-integer programming (MIP); to enhance scalability, we integrate complementary constrained (CC) optimization with a reduced MIP formulation, achieving up to a 17-fold reduction in runtime by approximating the verifiable region in a principled way; we provide a theoretical analysis characterizing the properties of the verifiable balls obtained through LEVIS-; and we validate our approach across diverse applications, including electrical power flow regression and image classification, demonstrating performance improvements and visualizing the geometric properties of the verifiable region. 

Abstract While whistler‐mode waves are generated by injected anisotropic electrons on the nightside, the observed day‐night asymmetry of wave distributions raises an intriguing question about their generation on the dayside. In this study, we evaluate the distributions of whistler‐mode wave amplitudes and electrons as a function of distance from the magnetopause (MP) on the dayside from 6 to 18 hr in magnetic local time (MLT) within ±18° of magnetic latitude using the Time History of Events and Macroscale Interaction During Substorms measurements from June 2010 to August 2018. Specifically, under different levels of solar wind dynamic pressure and geomagnetic index, we conduct a statistical analysis to examine whistler‐mode wave amplitude, as well as anisotropy and phase space density (PSD) of source electrons across 1–20 keV energies, which potentially provide a source of free energy for wave generation. In coordinates relative to the MP, we find that lower‐band (0.05–0.5f_ce) waves occur much closer to the MP than upper‐band (0.5–0.8f_ce) waves, wheref_ceis electron cyclotron frequency. Our statistical results reveal that strong waves are associated with high anisotropy and high PSD of source electrons near the equator, indicating a preferred region for local wave generation on the dayside. Over 10–14 hr in MLT, as latitude increases, electron anisotropy decreases, while whistler‐mode wave amplitudes increase, suggesting that wave propagation from the equator to higher latitudes, along with amplification along the propagation path, is necessary to explain the observed waves on the dayside. 

Abstract We evaluate the diffusive and nonlinear scattering of ring current protons by electromagnetic ion cyclotron (EMIC) waves in the Earth's inner magnetosphere using test particle simulations. EMIC waves are commonly observed inside and outside the plasmasphere with wave amplitudes ranging from 100 pT to several nT. Field‐aligned EMIC waves can scatter 1 keV–1 MeV protons counter‐streaming with respect to the waves through first order cyclotron resonance. Through the analyses of the proton equatorial pitch angle variations along the field line, our simulations reveal the typical interaction features including quasilinear diffusion for small wave amplitudes, phase trapping and bunching at intermediate and large pitch angles, anomalous phase trapping and positive phase bunching at small pitch angles, and non‐resonant scattering at pitch angles and energies outside the resonance regime. Using different wave amplitudes from 100 pT to 5 nT, we compared the modeling results of proton equatorial pitch angle variations between quasilinear and test particle simulations, and between diffusive scattering and advective effects. For monochromatic He‐band EMIC waves atL = 5, the interaction between protons and EMIC waves with amplitudes below 500 pT could be described as a diffusive process and quantified by quasilinear theory; nonlinear interactions and advection effects become important for wave amplitudes larger than 1 nT. The interactions between EMIC waves and ring current protons are analogous to the interactions between whistler‐mode chorus waves and radiation belt electrons described in previous studies, despite the quantitative differences in the wave amplitude threshold of quasilinear diffusion applicability. 

Abstract During geomagnetic storms relativistic outer radiation belt electron flux exhibits large variations on rapid time scales of minutes to days. Many competing acceleration and loss processes contribute to the dynamic variability of the radiation belts; however, distinguishing the relative contribution of each mechanism remains a major challenge as they often occur simultaneously and over a wide range of spatiotemporal scales. In this study, we develop a new comprehensive model for storm‐time radiation belt dynamics by incorporating electron wave‐particle interactions with parallel propagating whistler mode waves into our global test‐particle model of the outer belt. Electron trajectories are evolved through the electromagnetic fields generated from the Multiscale Atmosphere‐Geospace Environment (MAGE) global geospace model. Pitch angle scattering and energization of the test particles are derived from analytical expressions for quasi‐linear diffusion coefficients that depend directly on the magnetic field and density from the magnetosphere simulation. Using a study of the 17 March 2013 geomagnetic storm, we demonstrate that resonance with lower band chorus waves can produce rapid relativistic flux enhancements during the main phase of the storm. While electron loss from the outer radiation belt is dominated by loss through the magnetopause, wave‐particle interactions drive significant atmospheric precipitation. We also show that the storm‐time magnetic field and cold plasma density evolution produces strong, local variations of the magnitude and energy of the wave‐particle interactions and is critical to fully capturing the dynamic variability of the radiation belts caused by wave‐particle interactions.