Generative Modeling by Transport: Diffusions, Flows, and Interpolants

From Theory to Real

Ziseok Lee, Wooyeol Lee, Minyeong Hwang, Eunji Jeong, Donghwan Lee, Kyungsu Kim

2025.06.03 ~

Abstract

Diffusion models have emerged as a powerful class of generative models, achieving state-of-the-art performance in various tasks such as image synthesis, text-to-image generation, and video generation. This study provides a comprehensive overview of diffusion models, covering their theoretical foundations, practical implementations, and applications across different domains.

Study Plan

Part 1: Mathematical Foundation

• Stochastic processes, Brownian motion, Itô calculus
• SDEs and Fokker–Planck equations (forward & backward)
• Girsanov's Theorem: Change of Measure in Stochastic Processes

Part 2: Diffusion Model

• Score matching (denoising, score estimation)
• DDPM (Ho et al.), Score-based models (Song et al.)
• Sampling in Diffusion Models: Reverse-time SDEs and ODEs

Part 3: Stochastic Interpolant

• Consistency models, Flow matching, and Stochastic Interpolants

Part 4: Measure Transport

• Schrödinger Bridges, Optimal Transport, and Diffusion Models
• Diffusion bridges, functional diffusion processes

Application 1: Text-to-Image Generation

• Text-to-image (DALLE, Imagen, SD)
• Text-to-3D, multimodal alignment with LLMs
• Advanced Sampling: Latent diffusion, Conditional Generation, CFG, Manifold Hypothesis (CFG++)

Application 2: Molecule Generation

• Protein folding/design (e.g., AlphaFold, RFdiffusion)
• DNA/RNA generation

Application 3: Diffusion Language Model

• Diffusion + LLM

Application 4: Robotic Control

• Diffusion + RL (diffusion policy)

Implementations

• Train a DDPM on CIFAR-10
• Train a Flow Model on CIFAR-10

References

Books and Research Papers

• Understanding Diffusion Models: A Unified Perspective, 2022
• Stochastic Interpolants: A Unifying Framework for Flows and Diffusions, 2023
• Continuous-Time Functional Diffusion Processes, 2023
• Statistical Optimal Transport, 2024
• Tutorial on Diffusion Models for Imaging and Vision, 2024
• A Unified Approach to Analysis and Design of Denoising Markov Models, 2025
• Lecture Notes on Applied Mathematics, 2009
• Optimal Transport for Machine Learners, 2025
• Applied Stochastic Differential Equations, 2019
• Flow Matching Guide and Code, 2024

Lectures

• MIT Computer Science Class 6.S184: Generative AI with Stochastic Differential Equations
• An Introduction to Flow Matching and Diffusion Models