Generative Flows

Generative models that transport a simple distribution to a complex target.

Part I · Two Tracks

The Transport Problem Normalizing flows, coupling layers, and the birth of explicit generative transport

The Continuous Limit Neural ODEs, continuous normalizing flows, and the simulation bottleneck

The Denoising Perspective Score matching, diffusion models, and the probability flow ODE

Flow Matching and Stochastic Interpolants Convergence of two tracks — simulation-free training and the unification moment

Part II · Transport Theory

Optimal Transport and Path Geometry Monge, Kantorovich, Wasserstein, the Benamou-Brenier formula, and the geometry of straight paths

Schrödinger Bridges and Entropic Transport Entropy-regularized OT, iterative proportional fitting, bridge matching, and diffusion bridges

Stochastic Flows and the SDE–ODE Duality When and why stochasticity helps — Langevin correctors, drift-based generation, and the design of noise

Part III · The Modern Framework

The Generator Matching Framework Every method in one theorem — Markov generators and Bregman losses

Beyond Euclidean Space Discrete flows via CTMCs, Riemannian flow matching, and flows on the simplex

Joint Generation on Product Spaces Co-generating continuous and discrete data — the current frontier

Symmetry, Conservation Laws, and Inductive Biases Equivariant architectures, divergence-free fields, and physics-informed constraints

Training Objectives, Losses, and Parameterizations MLE, regression losses, score matching, energy-based training, and practical recipes

Part IV · Sampling, Correction, and Control

Fast Sampling and Flow Maps Consistency models, stochastic flow maps, distillation, and the race to one step

Feynman-Kac Correctors and Sequential Monte Carlo Importance sampling, path integrals, and correcting imperfect models post-hoc

Guided and Conditional Generation Classifier guidance, classifier-free guidance, reward steering, and inverse problems