@article{21504,
  abstract     = {Selecting an appropriate divergence measure is a critical aspect of machine learning, as it directly impacts model performance. Among the most widely used, we find the Kullback–Leibler (KL) divergence, originally introduced in kinetic theory as a measure of relative entropy between probability distributions. Just as in machine learning, the ability to quantify the proximity of probability distributions plays a central role in kinetic theory. In this paper, we present a comparative review of divergence measures rooted in kinetic theory, highlighting their theoretical foundations and exploring their potential applications in machine learning and artificial intelligence.},
  author       = {Auricchio, Gennaro and Brigati, Giovanni and Giudici, Paolo and Toscani, Giuseppe},
  issn         = {1793-6314},
  journal      = {Mathematical Models and Methods in Applied Sciences},
  publisher    = {World Scientific Publishing},
  title        = {{From kinetic theory to AI: A rediscovery of high-dimensional divergences and their properties}},
  doi          = {10.1142/S0218202526410010},
  year         = {2026},
}

@article{19565,
  abstract     = {Measuring distances in a multidimensional setting is a challenging problem, which appears in many fields of science and engineering. In this paper, to measure the distance between two multivariate distributions, we introduce a new measure of discrepancy which is scale invariant and which, in the case of two independent copies of the same distribution, and after normalization, coincides with the scaling invariant multidimensional version of the Gini index recently proposed in [P. Giudici, E. Raffinetti and G. Toscani, Measuring multidimensional inequality: A new proposal based on the Fourier transform, preprint (2024), arXiv:2401.14012 ]. A byproduct of the analysis is an easy-to-handle discrepancy metric, obtained by application of the theory to a pair of Gaussian multidimensional densities. The obtained metric does improve the standard metrics, based on the mean squared error, as it is scale invariant. The importance of this theoretical finding is illustrated by means of a real problem that concerns measuring the importance of Environmental, Social and Governance factors for the growth of small and medium enterprises. },
  author       = {Auricchio, Gennaro and Brigati, Giovanni and Giudici, Paolo and Toscani, Giuseppe},
  issn         = {1793-6314},
  journal      = {Mathematical Models and Methods in Applied Sciences},
  number       = {5},
  pages        = {1267--1296},
  publisher    = {World Scientific Publishing},
  title        = {{Multivariate Gini-type discrepancies}},
  doi          = {10.1142/s0218202525500174},
  volume       = {35},
  year         = {2025},
}

@article{10575,
  abstract     = {The choice of the boundary conditions in mechanical problems has to reflect the interaction of the considered material with the surface. Still the assumption of the no-slip condition is preferred in order to avoid boundary terms in the analysis and slipping effects are usually overlooked. Besides the “static slip models”, there are phenomena that are not accurately described by them, e.g. at the moment when the slip changes rapidly, the wall shear stress and the slip can exhibit a sudden overshoot and subsequent relaxation. When these effects become significant, the so-called dynamic slip phenomenon occurs. We develop a mathematical analysis of Navier–Stokes-like problems with a dynamic slip boundary condition, which requires a proper generalization of the Gelfand triplet and the corresponding function space setting.},
  author       = {Abbatiello, Anna and Bulíček, Miroslav and Maringová, Erika},
  issn         = {1793-6314},
  journal      = {Mathematical Models and Methods in Applied Sciences},
  number       = {11},
  pages        = {2165--2212},
  publisher    = {World Scientific Publishing},
  title        = {{On the dynamic slip boundary condition for Navier-Stokes-like problems}},
  doi          = {10.1142/S0218202521500470},
  volume       = {31},
  year         = {2021},
}

@article{10005,
  abstract     = {We study systems of nonlinear partial differential equations of parabolic type, in which the elliptic operator is replaced by the first-order divergence operator acting on a flux function, which is related to the spatial gradient of the unknown through an additional implicit equation. This setting, broad enough in terms of applications, significantly expands the paradigm of nonlinear parabolic problems. Formulating four conditions concerning the form of the implicit equation, we first show that these conditions describe a maximal monotone p-coercive graph. We then establish the global-in-time and large-data existence of a (weak) solution and its uniqueness. To this end, we adopt and significantly generalize Minty’s method of monotone mappings. A unified theory, containing several novel tools, is developed in a way to be tractable from the point of view of numerical approximations.},
  author       = {Bulíček, Miroslav and Maringová, Erika and Málek, Josef},
  issn         = {1793-6314},
  journal      = {Mathematical Models and Methods in Applied Sciences},
  keywords     = {Nonlinear parabolic systems, implicit constitutive theory, weak solutions, existence, uniqueness},
  number       = {09},
  publisher    = {World Scientific Publishing},
  title        = {{On nonlinear problems of parabolic type with implicit constitutive equations involving flux}},
  doi          = {10.1142/S0218202521500457},
  volume       = {31},
  year         = {2021},
}