{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,19]],"date-time":"2025-11-19T07:13:19Z","timestamp":1763536399309,"version":"build-2065373602"},"reference-count":25,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2025,6,30]],"date-time":"2025-06-30T00:00:00Z","timestamp":1751241600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Universidade Estadual de Santa Cruz (UESC)"},{"name":"Funda\u00e7\u00e3o de Ampararo \u00e0 Pesquisa do Estado da Bahia (FAPESB)"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This work presents a comprehensive mathematical framework for symmetrized neural network operators operating under the paradigm of fractional calculus. By introducing a perturbed hyperbolic tangent activation, we construct a family of localized, symmetric, and positive kernel-like densities, which form the analytical backbone for three classes of multivariate operators: quasi-interpolation, Kantorovich-type, and quadrature-type. A central theoretical contribution is the derivation of the Voronovskaya\u2013Santos\u2013Sales Theorem, which extends classical asymptotic expansions to the fractional domain, providing rigorous error bounds and normalized remainder terms governed by Caputo derivatives. The operators exhibit key properties such as partition of unity, exponential decay, and scaling invariance, which are essential for stable and accurate approximations in high-dimensional settings and systems governed by nonlocal dynamics. The theoretical framework is thoroughly validated through applications in signal processing and fractional fluid dynamics, including the formulation of nonlocal viscous models and fractional Navier\u2013Stokes equations with memory effects. Numerical experiments demonstrate a relative error reduction of up to 92.5% when compared to classical quasi-interpolation operators, with observed convergence rates reaching On\u22121.5 under Caputo derivatives, using parameters \u03bb=3.5, q=1.8, and n=100. This synergy between neural operator theory, asymptotic analysis, and fractional calculus not only advances the theoretical landscape of function approximation but also provides practical computational tools for addressing complex physical systems characterized by long-range interactions and anomalous diffusion.<\/jats:p>","DOI":"10.3390\/axioms14070510","type":"journal-article","created":{"date-parts":[[2025,6,30]],"date-time":"2025-06-30T13:06:17Z","timestamp":1751288777000},"page":"510","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Symmetrized Neural Network Operators in Fractional Calculus: Caputo Derivatives, Asymptotic Analysis, and the Voronovskaya\u2013Santos\u2013Sales Theorem"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9482-1998","authenticated-orcid":false,"given":"R\u00f4mulo Damasclin Chaves dos","family":"Santos","sequence":"first","affiliation":[{"name":"Department of Exact Sciences, Postgraduate Program in Computational Modeling Santa Cruz State University, Ilh\u00e9us 45662-900, Brazil"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1992-3748","authenticated-orcid":false,"given":"Jorge Henrique de Oliveira","family":"Sales","sequence":"additional","affiliation":[{"name":"Department of Exact Sciences, Postgraduate Program in Computational Modeling Santa Cruz State University, Ilh\u00e9us 45662-900, Brazil"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9067-5815","authenticated-orcid":false,"given":"Gislan Silveira","family":"Santos","sequence":"additional","affiliation":[{"name":"Department of Exact Sciences, Postgraduate Program in Computational Modeling Santa Cruz State University, Ilh\u00e9us 45662-900, Brazil"}]}],"member":"1968","published-online":{"date-parts":[[2025,6,30]]},"reference":[{"key":"ref_1","unstructured":"Alonso, N.I. 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