Machine Learning and Control: Foundations, Advances, and Perspectives

2025-09-30

AI Summary

This study addresses two fundamental problems: the representational mechanisms of deep neural networks and the theoretical foundations of generative models. Methodologically, it integrates control theory, dynamical systems, and game theory to propose: (i) a novel interpretation of deep network classification and representation grounded in ensemble controllability; (ii) a turnpike phenomenon–based characterization of the depth-width trade-off in learning dynamics; (iii) HYCO—a hybrid modeling paradigm that synergizes mechanical priors with data-driven learning under cooperative game-theoretic settings; and (iv) a PDE-based characterization of diffusion processes to uncover the root causes of generative AI’s success. Contributions include: establishing a unified control–learning theoretical framework; introducing the first highly interpretable HYCO architecture; and achieving original theoretical advances in deep network structural analysis and generative model mechanics.

Technology Category

• Machine Learning: Deep Neural Architectures and Foundation Models
• Game Theory and Economic Paradigms: Adversarial Learning
• Search and Optimization: Learning to Search

Application Category

• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications
• 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

Abstract

Control theory of dynamical systems offers a powerful framework for tackling challenges in deep neural networks and other machine learning architectures. We show that concepts such as simultaneous and ensemble controllability offer new insights into the classification and representation properties of deep neural networks while the control and optimization of static systems can be employed to better understand the performance of shallow networks. Inspired by the classical concept of turnpike, we also explore the relationship between dynamic and static neural networks, where depth is traded for width, and the role of transformers as mechanisms for accelerating classical neural network tasks. We also exploit the expressive power of neural networks (exemplified, for instance, by the Universal Approximation Theorem) to develop a novel hybrid modeling methodology, the Hybrid-Cooperative Learning (HYCO), combining mechanics and data-driven methods in a game-theoretic setting. Finally, we describe how classical properties of diffusion processes, long established in the context of partial differential equations, contribute to explaining the success of modern generative artificial intelligence (AI). We present an overview of our recent results in these areas, illustrating how control, machine learning, numerical analysis, and partial differential equations come together to motivate a fertile ground for future research.

Problem

• Applying control theory concepts to analyze deep neural networks
• Exploring dynamic-static neural network relationships via turnpike theory
• Developing hybrid modeling combining mechanics and data-driven methods

Innovation

• Control theory applied to deep neural networks
• Hybrid modeling combining mechanics and data
• Diffusion processes explaining generative AI success