Neural Networks: Mathematical Foundations and Applications

• Grant number: 24/17594-5
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Start date: November 01, 2024
• End date: October 31, 2025
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Analysis
• Principal Investigator: Murilo Rodolfo Cândido
• Grantee: Gabriel Augusto Martins
• Host Institution: Faculdade de Ciências e Tecnologia (FCT). Universidade Estadual Paulista (UNESP). Campus de Presidente Prudente. Presidente Prudente , SP, Brazil
• Associated research grant: 23/06076-0 - Averaging based methods for dectecting invariant compact sets, AP.JP

Abstract

The present project aims to explore the theoretical foundations of artificial neural networks, using a rigorous pure mathematics approach to explain and formalize the fundamental theorems that support their success in modern applications. We propose a detailed analysis of important mathematical results, such as the Universal Approximation Theorem, examining its conditions, limitations, and possible extensions. Additionally, we plan to apply these theoretical principles to solve real-world problems, such as image classification, showing the connection between the mathematical theory and its applications in machine learning. This study aims to contribute to the progress of knowledge at the intersection of pure mathematics and artificial intelligence, providing insights that may guide future research.