Journal of Samara State Technical University, Ser. Physical and Mathematical SciencesJournal of Samara State Technical University, Ser. Physical and Mathematical SciencesВестник Самарского государственного технического университета. Серия «Физико-математические науки»1991-86152310-7081Samara State Technical University63456610.14498/vsgtu2105ENXAZEMathematical Modeling, Numerical Methods and Software ComplexesМатематическое моделирование, численные методы и комплексы программResearch ArticleTriply periodic surface description using Laplace–Beltrami operator and a statistical machine learning modelОписание трижды периодических поверхностей с помощью оператора Лапласа–Бельтрами и статистической модели машинного обученияhttps://orcid.org/0000-0001-5573-662XSmolkovMikhail I.СмольковМихаил Игоревич
Russian Federation

Postgraduate Research Student; Junior Researcher; International Research Center for Theoretical Materials Science

аспирант; младший научный сотрудник; международный научно-исследовательский центр по теоретическому материаловедению

m.smolkov97@gmail.comhttps://www.mathnet.ru/person227410Samara State Technical UniversityСамарский государственный технический университет200520252911581732407202406032025Copyright ©; 2025, Authors; Samara State Technical University (Compilation, Design, and Layout)Copyright ©; 2025, Авторский коллектив; Самарский государственный технический университет (составление, дизайн, макет)2025Authors; Samara State Technical University (Compilation, Design, and Layout)Авторский коллектив; Самарский государственный технический университет (составление, дизайн, макет)https://creativecommons.org/licenses/by/4.0https://journals.eco-vector.com/1991-8615/article/view/634566

Triply periodic surfaces (TPS) and their minimal analogs (TPMS) are currently widely used in various fields, including mechanics, biomechanics, aerodynamics, hydrodynamics, and radiophysics. In this context, the problem of establishing correlations between the topological and geometric properties of surfaces and their physical characteristics arises. To address this problem, it is necessary to introduce a measure of similarity between surfaces with different topological and geometric features. This work focuses on describing TPS and TPMS in terms of a specific metric space of descriptors. The problem is solved using the mathematical framework of image recognition theory. A descriptor is constructed based on a set of eigenvectors and eigenvalues of the Beltrami–Laplace operator and a joint Bayesian model. A metric based on a probabilistic measure of surface similarity is introduced in the descriptor space. The effectiveness of the method developed in this work has been tested on 51 surfaces of class P. The accuracy of predicting the surface type is 92.8 %. The developed machine learning model enables the determination of whether a given surface belongs to the class of P-surfaces.

Трижды периодические поверхности (ТПП) и их минимальные аналоги (ТПМП) в настоящее время активно применяются в различных областях, таких как механика, биомеханика, аэродинамика, гидродинамика и радиофизика. В связи с этим возникает задача установления корреляций между тополого-геометрическими свойствами поверхностей и их физическими характеристиками. Для решения данной задачи необходимо ввести меру сходства между поверхностями, обладающими различными тополого-геометрическими свойствами. Настоящая работа посвящена описанию ТПП и ТПМП в терминах метрического пространства дескрипторов. Решение задачи осуществляется с использованием математического аппарата теории распознавания изображений. Построен дескриптор на основе совокупности собственных векторов и собственных значений оператора Бельтрами–Лапласа, а также совместной байесовской модели. В пространстве дескрипторов введена метрика, основанная на вероятностной мере сходства поверхностей. Работоспособность разработанного метода проверена на 51 поверхности класса P. Точность предсказания типа поверхности составила 92.8 %. Разработанная модель машинного обучения позволяет определить принадлежность произвольной поверхности к классу P-поверхностей.

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