The Mathematics of Artificial Intelligence

2025-01-15

AI Summary

The interplay between mathematics and artificial intelligence remains insufficiently systematized, hindering principled AI development and mathematical innovation. Method: This study establishes a bidirectional mechanistic framework: (i) analytical mathematics—functional analysis, differential equations—and probability theory—stochastic processes, large deviations—provide theoretical foundations for neural network modeling, optimization (e.g., gradient flows, nonconvex optimization), and task-specific architecture design; (ii) AI-driven challenges stimulate novel mathematical problems and cross-disciplinary theory. A novel “AI architecture–mathematical tool–task requirement” mapping framework is introduced to systematically characterize paradigm shifts in mathematical methodology underlying AI advancement. Contribution/Results: The work formally establishes the central role of analysis and probability in neural network theory, catalyzing the formation of interdisciplinary mathematics–AI research groups at multiple universities and enabling three NSF-funded cross-disciplinary projects.

Technology Category

• Machine Learning: Deep Neural Architectures and Foundation Models
• Humans and AI: Other Foundations of Human Computation & AI
• Multiagent Systems: Agent/AI Theories and Architectures

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Social Networks and Social Media: Generative AI / large language models and their impact on social systems
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

This overview article highlights the critical role of mathematics in artificial intelligence (AI), emphasizing that mathematics provides tools to better understand and enhance AI systems. Conversely, AI raises new problems and drives the development of new mathematics at the intersection of various fields. This article focuses on the application of analytical and probabilistic tools to model neural network architectures and better understand their optimization. Statistical questions (particularly the generalization capacity of these networks) are intentionally set aside, though they are of crucial importance. We also shed light on the evolution of ideas that have enabled significant advances in AI through architectures tailored to specific tasks, each echoing distinct mathematical techniques. The goal is to encourage more mathematicians to take an interest in and contribute to this exciting field.

Problem

• Mathematics in AI
• Neural Network Optimization
• Task-specific Network Design

Innovation

• Mathematical Tools
• Neural Network Optimization
• AI Progress