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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 Interchange instability is known to drive fast radial transport of electrons and ions in Jupiter's inner and middle magnetosphere. In this study, we conduct a statistical survey to evaluate the properties of energetic particles and plasma waves during interchange events using Juno data from 2016 to 2023. We present representative examples of interchange events followed by a statistical analysis of the spatial distribution, duration and spatial extent. Our survey indicates that interchange instability is predominant atM‐shells from 6 to 26, peaking near 17 with an average duration of minutes and a correspondingM‐shell width of <∼0.05. During interchange events, the associated plasma waves, such as whistler‐mode, Z‐mode, and electron cyclotron harmonic waves exhibit a distinct preferential location. These findings provide valuable insights into particle transport and the source region of plasma waves in the Jovian magnetosphere, as well as in other magnetized planets within and beyond our solar system. 

Abstract Energetic electron precipitation (EEP) from the radiation belts into Earth's atmosphere leads to several profound effects (e.g., enhancement of ionospheric conductivity, possible acceleration of ozone destruction processes). An accurate quantification of the energy input and ionization due to EEP is still lacking due to instrument limitations of low‐Earth‐orbit satellites capable of detecting EEP. The deployment of the Electron Losses and Fields InvestigatioN (ELFIN) CubeSats marks a new era of observations of EEP with an improved pitch‐angle (0°–180°) and energy (50 keV–6 MeV) resolution. Here, we focus on the EEP recorded by ELFIN coincident with electromagnetic ion cyclotron (EMIC) waves, which play a major role in radiation belt electron losses. The EMIC‐driven EEP (∼200 keV–∼2 MeV) exhibits a pitch‐angle distribution (PAD) that flattens with increasing energy, indicating more efficient high‐energy precipitation. Leveraging the combination of unique electron measurements from ELFIN and a comprehensive ionization model known as Boulder Electron Radiation to Ionization (BERI), we quantify the energy input of EMIC‐driven precipitation (on average, ∼3.3 × 10⁻² erg/cm²/s), identify its location (any longitude, 50°–70° latitude), and provide the expected range of ion‐electron production rate (on average, 100–200 pairs/cm³/s), peaking in the mesosphere—a region often overlooked. Our findings are crucial for improving our understanding of the magnetosphere‐ionosphere‐atmosphere system as they accurately specify the contribution of EMIC‐driven EEP, which serves as a crucial input to state‐of‐the‐art atmospheric models (e.g., WACCM) to quantify the accurate impact of EMIC waves on both the atmospheric chemistry and dynamics. 

Abstract We present statistical distributions of whistler‐mode chorus and hiss waves at frequencies ranging from the local proton gyrofrequency to the equatorial electron gyrofrequency (f_ce,eq) in Jupiter's magnetosphere based on Juno measurements. The chorus wave power spectral densities usually follow thef_ce,eqvariation with major wave power concentrated in the 0.05f_ce,eq–f_ce,eqfrequency range. The hiss wave frequencies are less dependent onf_ce,eqvariation than chorus with major power concentrated below 0.05f_ce,eq, showing a separation from chorus atM < 10. Our survey indicates that chorus waves are mainly observed at 5.5 < M < 13 from the magnetic equator to 20° latitude, consistent with local wave generation near the equator and damping effects. The hiss wave powers extend to 50° latitude, suggesting longer wave propagation paths without attenuation. Our survey also includes the whistler‐mode waves at high latitudes which may originate from the Io footprint, auroral hiss, or propagating hiss waves reflected to highMshells.