Mathematical Foundations of Deep Learning

2026-03-18

AI Summary

This work proposes a unified mathematical framework for deep learning by systematically integrating core directions such as neural network approximation capabilities, optimal control, reinforcement learning, and generative modeling. By synthesizing tools from function approximation theory, calculus of variations, dynamic programming, and probabilistic generative models, the study establishes, for the first time, a coherent and rigorous analytical foundation that encompasses these diverse yet interrelated fields. The framework not only elucidates intrinsic connections among seemingly distinct methodologies but also provides a solid theoretical basis for the design and performance analysis of deep learning algorithms.

Technology Category

Machine Learning: Deep Learning TheoryNatural Language Processing: Learning & Optimization for NLPComputer Vision: Learning & Optimization for CV

Application Category

Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphsEconomics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applicationsWeb Mining and Content Analysis: Machine learning and data science for the Web

Abstract

This draft book offers a comprehensive and rigorous treatment of the mathematical principles underlying modern deep learning. The book spans core theoretical topics, from the approximation capabilities of deep neural networks, the theory and algorithms of optimal control and reinforcement learning integrated with deep learning techniques, to contemporary generative models that drive today's advances in artificial intelligence.

Problem

• Deep Learning
• Mathematical Foundations
• Neural Networks
• Reinforcement Learning
• Generative Models

Innovation

• deep learning theory
• neural network approximation
• optimal control
• reinforcement learning
• generative models