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1. Mean field limits of particle-based stochastic reaction-drift-diffusion models ^*

   https://doi.org/10.1088/1361-6544/ad8fea  

   Heldman, M; Isaacson, S A; Liu, Q; Spiliopoulos, K (January 2025, Nonlinearity)

   Abstract We consider particle-based stochastic reaction-drift-diffusion models where particles move via diffusion and drift induced by one- and two-body potential interactions. The dynamics of the particles are formulated as measure-valued stochastic processes (MVSPs), which describe the evolution of the singular, stochastic concentration fields of each chemical species. The mean field large population limit of such models is derived and proven, giving coarse-grained deterministic partial integro-differential equations (PIDEs) for the limiting deterministic concentration fields’ dynamics. We generalize previous studies on the mean field limit of models involving only diffusive motion, with care to formulating the MVSP representation to ensure detailed balance of reversible reactions in the presence of potentials. Our work illustrates the more general set of PIDEs that arise in the mean field limit, demonstrating that the limiting macroscopic reactive interaction terms for reversible reactions obtain additional nonlinear concentration-dependent coefficients compared to the purely diffusive case. Numerical studies are presented which illustrate that two-body repulsive potential interactions can have a significant impact on the reaction dynamics, and also demonstrate the empirical numerical convergence of solutions to the PBSRDD model to the derived mean field PIDEs as the population size increases. 

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2. Self-Improving inequalities for bounded weak solutions to nonlocal double phase equations

   https://doi.org/10.3934/cpaa.2021174  

   Scott, James M.; Mengesha, Tadele (January 2021, Communications on Pure & Applied Analysis)

   We prove higher Sobolev regularity for bounded weak solutions to a class of nonlinear nonlocal integro-differential equations. The leading operator exhibits nonuniform growth, switching between two different fractional elliptic "phases" that are determined by the zero set of a modulating coefficient. Solutions are shown to improve both in integrability and differentiability. These results apply to operators with rough kernels and modulating coefficients. To obtain these results we adapt a particular fractional version of the Gehring lemma developed by Kuusi, Mingione, and Sire in their work "Nonlocal self-improving properties" Analysis & PDE, 8(1):57–114 for the specific nonlinear setting under investigation in this manuscript. 

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3. Statistical and Machine Learning Approaches for Electrical Energy Forecasting

   https://doi.org/10.1002/widm.70033  

   Machado, Solange; Zhu, Xingquan (September 2025, WIREs Data Mining and Knowledge Discovery)

   ABSTRACT With renewable energy being aggressively integrated into the grid, energy supplies are becoming vulnerable to weather and the environment, and are often incapable of meeting population demands at a large scale if not accurately predicted for energy planning. Understanding consumers' power demands ahead of time and the influences of weather on consumption and generation can help producers generate effective power management plans to support the target demand. In addition to the high correlation with the environment, consumers' behaviors also cause non‐stationary characteristics of energy data, which is the main challenge for energy prediction. In this survey, we perform a review of the literature on prediction methods in the energy field. So far, most of the available research encompasses one type of generation or consumption. There is no research approaching prediction in the energy sector as a whole and its correlated features. We propose to address the energy prediction challenges from both consumption and generation sides, encompassing techniques from statistical to machine learning techniques. We also summarize the work related to energy prediction, electricity measurements, challenges related to energy consumption and generation, energy forecasting methods, and real‐world energy forecasting resources, such as datasets and software solutions for energy prediction. This article is categorized under:Application Areas > Industry Specific ApplicationsTechnologies > PredictionTechnologies > Machine Learning 

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4. Estimates for Dirichlet-to-Neumann Maps as Integro-differential Operators

   https://doi.org/10.1007/s11118-019-09776-w  

   Guillen, Nestor; Kitagawa, Jun; Schwab, Russell W. (April 2019, Potential analysis)

   Some linear integro-differential operators have old and classical representations as the Dirichlet-to-Neumann operators for linear elliptic equations, such as the 1/2-Laplacian or the generator of the boundary process of a reflected diffusion. In this work, we make some extensions of this theory to the case of a nonlinear Dirichlet-to-Neumann mapping that is constructed using a solution to a fully nonlinear elliptic equation in a given domain, mapping Dirichlet data to its normal derivative of the resulting solution. Here we begin the process of giving detailed information about the Lévy measures that will result from the integro-differential representation of the Dirichlet-to-Neumann mapping. We provide new results about both linear and nonlinear Dirichlet-to-Neumann mappings. Information about the Lévy measures is important if one hopes to use recent advancements of the integro-differential theory to study problems involving Dirichlet-to-Neumann mappings. 

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5. The Growth of Ice Crystals Formed from Frozen Solution Droplets: Laboratory Measurements and Power-Law Parameterization

   https://doi.org/10.1175/JAS-D-24-0182.1  

   Rui, Dixuan; Pokrifka, Gwenore F; Moyle, Alfred M; Harrington, Jerry Y (April 2025, Journal of the Atmospheric Sciences)

   Abstract All cloud and climate models assume ice crystals grow as if they were formed from pure water, even though cloud and haze droplets are solutions. The freezing process of a solution droplet is different than that of a pure water droplet, as shown in prior work. This difference can potentially affect the particle’s subsequent growth as an ice crystal. We present measurements of ice crystal growth from frozen sodium chloride (NaCl) solution droplets in the button electrode levitation diffusion chamber at temperatures between −61° and −40°C. Measured scattering patterns show that concentrated solution droplets remain unfrozen with classical scattering fringes until the droplets freeze. Upon freezing, the scattering patterns become complex within 0.1 s, which is in contrast with frozen pure water particles that retain liquid-like scattering patterns for about a minute. We show that after freezing, solution particles initially grow as spherical-like crystals and then transition to faster growth indicative of a morphological transformation. The measurements indicate that ice formed from solution droplets grows differently and has higher growth rates than ice formed from pure water droplets. We use these results to develop a power-law-based parameterization that captures the supersaturation and mass dependencies. 

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