Score-based Neural Ordinary Differential Equations and Normalizing Flow for Mean Field Control

Overview

Explore a novel approach to Mean Field Control using score-based neural ODEs and normalizing flows, with applications in generative models, probability flow matching, and Wasserstein proximal operators.

Syllabus

• Introduction
Overview of Mean Field Control
Introduction to Neural Ordinary Differential Equations (Neural ODEs)
Basics of Normalizing Flows
Course Objectives and Structure
• Review of Mathematical Foundations
Differential Equations and Dynamical Systems
Probability Theory and Stochastic Processes
Optimal Transport and Wasserstein Distances
• Neural Ordinary Differential Equations (Neural ODEs)
Mechanics of ODE-Based Models
Advantages of Neural ODEs in Control
Applications in Sequential Data Generation
• Score-Based Models and Generative Modeling
Introduction to Score-Based Models
Connection with Existing Generative Models
Probabilistic Interpretation and Estimation
• Normalizing Flows
Concept and Mechanics of Normalizing Flows
Flow-Based Generative Models
Training and Inference Techniques
• Score-Based Neural ODEs
Integration of Score Functions with Neural ODEs
Framework for Score-Based Dynamics
Case Studies and Practical Applications
• Mean Field Control (MFC)
Introduction to Mean Field Games and Control
Role of Neural ODEs in MFC
Implementation Strategies for MFC
• Probability Flow Matching
The Concept of Probability Flow
Techniques for Probability Flow Matching using Neural ODEs
• Wasserstein Proximal Operators
Introduction to Wasserstein Geometry
Construction of Wasserstein Proximal Operators
Applications in Optimization and Control
• Applications and Case Studies
Real-World Use Cases in Generative Models
Advanced Applications in Robotics and System Control
Exploration of Current Research Frontiers
• Practical Implementation
Software Tools and Platforms
Building and Training Neural ODE Models
Evaluating and Validating Models
• Conclusion
Summary and Key Takeaways
Future Directions in Research and Application
Final Project Guidelines
• Resources and Further Reading
Recommended Books and Papers
Online Courses and Lectures
Software Libraries and Tools

Subjects

Computer Science