AI-Powered Approaches for Hypersurface Reconstruction in Multidimensional Spaces

Kostadin Yotov, Emil Hadzhikolev (), Stanka Hadzhikoleva and Mariyan Milev
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Kostadin Yotov: Faculty of Mathematics and Informatics, University of Plovdiv Paisii Hilendarski, 236 Bulgaria Blvd., 4027 Plovdiv, Bulgaria
Emil Hadzhikolev: Faculty of Mathematics and Informatics, University of Plovdiv Paisii Hilendarski, 236 Bulgaria Blvd., 4027 Plovdiv, Bulgaria
Stanka Hadzhikoleva: Faculty of Mathematics and Informatics, University of Plovdiv Paisii Hilendarski, 236 Bulgaria Blvd., 4027 Plovdiv, Bulgaria
Mariyan Milev: Faculty of Economics and Business Administration, Sofia University St. Kliment Ohridski, 125 Tsarigradsko Shosse Blvd., Bl.3., 1113 Sofia, Bulgaria

Mathematics, 2024, vol. 12, issue 20, 1-30

Abstract: The present article explores the possibilities of using artificial neural networks to solve problems related to reconstructing complex geometric surfaces in Euclidean and pseudo-Euclidean spaces, examining various approaches and techniques for training the networks. The main focus is on the possibility of training a set of neural networks with information about the available surface points, which can then be used to predict and complete missing parts. A method is proposed for using separate neural networks that reconstruct surfaces in different spatial directions, employing various types of architectures, such as multilayer perceptrons, recursive networks, and feedforward networks. Experimental results show that artificial neural networks can successfully approximate both smooth surfaces and those containing singular points. The article presents the results with the smallest error, showcasing networks of different types, along with a technique for reconstructing geographic relief. A comparison is made between the results achieved by neural networks and those obtained using traditional surface approximation methods such as Bézier curves, k-nearest neighbors, principal component analysis, Markov random fields, conditional random fields, and convolutional neural networks.

Keywords: hypersurface reconstruction; approximation; artificial neural networks; Bézier curves; k-nearest neighbors; principal component analysis; Markov random fields; conditional random fields; convolutional neural networks (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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