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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 Whistler‐mode chorus waves have been widely examined in the Earth's magnetosphere since they play an important role in the dynamics of the radiation belts around the Earth and energetic electron precipitation into the upper atmosphere. In this study, we examine the relationship between the low‐energy (<800 eV) electrons and the linear instability of chorus waves through a statistical analysis of 12 years of in situ observations from the THEMIS mission. Our results show that 31% of all lower‐band chorus wave events observed near the magnetic equator (|MLAT| < 10°) are linearly unstable if the low‐energy electron density is considered, whereas only 4% are unstable when the low‐energy electrons are ignored. Additionally, we identify a critical transition in chorus wave properties, where emission types shift from predominantly electromagnetic ( < 0.03) to a mixture of electromagnetic and quasi‐electrostatic components ( ∼ 0.1) depending on the low‐energy electron density ratio. Moreover, an analysis of correlation coefficients () associated with temporal variations of wave amplitude reveals that the low‐energy electron density is the dominant factor (∼43% with  > 0.2) of chorus wave modulation, while the hot electron temperature (>800 eV) ratio and electron parallel beta contribute only 15% and 21%, respectively. These results emphasize the important contribution of the low‐energy electrons to the instability and modulations of chorus waves in the Earth's magnetosphere.