Unifying Generative Models with Path Integrals

We survey continuous-time generative modeling methods based on transporting a simple reference distribution to a data distribution via stochastic or deterministic dynamics. We present a unified framework in which diffusion models, score-based generative models, and flow matching are instances of learning a time-dependent vector field that induces a family of marginals $(ρ_t)_{t \in [0,1]}$ governed by continuity and Fokker-Planck equations. Such a unified theory is timely because these methods are converging methodologically, yet fragmented notation and competing derivations continue to obscure their shared structure and the practical tradeoffs governing sampling, stability, and computation. Within this framework, we (i) derive reverse-time sampling for diffusion and score-based models as controlled stochastic dynamics, (ii) show that the probability flow ODE yields identical marginals and connects diffusion to likelihood-based normalizing flows, and (iii) interpret flow matching as direct regression of the velocity field under a chosen interpolation, clarifying when it coincides with or differs from score-based training. We compare objectives, sampling schemes, and discretization errors under unified notation, discuss connections to Schrodinger bridges and entropic optimal transport, and summarize theoretical guarantees and open problems on approximation, stability, and scalability.

Modern generative modeling is dominated by transport from a noise prior to data. We propose an alternative paradigm in which generation is performed by a discrete stochastic dynamics that leaves the data distribution invariant, initialized from data-supported states rather than from noise. The framework can utilize any pretrained flow model. We develop two probability-preserving sampling mechanisms, a corrected Langevin dynamics with a Metropolis adjustment and a predictor-corrector flow, that operate directly on existing checkpoints. We validate the framework on a synthetic Swiss-roll target, ImageNet-256 and Oxford Flowers-102, where our samplers consistently improve over the original generation procedures.

We introduce a novel Wasserstein-1 ($W_1$) path-space divergence for stochastic and deterministic dynamics and establish a Wasserstein Uncertainty Propagation (WUP) theorem that bounds the $W_1$ distance between terminal distributions by the proposed divergence, equivalently characterized by a weighted $L^2$ discrepancy between the underlying drifts and the $W_1$ distance between their initial measures. A key ingredient is a probabilistic framework combining adjoint Feynman-Kac representations with synchronous coupling (and reflection coupling on bounded domains), yielding Wasserstein stability estimates beyond existing PDE- and Girsanov-based approaches. The framework accommodates time-varying and possibly degenerate diffusion coefficients, empirical and singular measures, and remains valid in the deterministic limit of flow matching. Unlike KL-based uncertainty quantification bounds, it does not require absolute continuity of path measures and therefore remains well-defined in singular settings. As consequences of the WUP theorem, we derive $W_1$ robustness and generalization bounds for score-based generative models and flow matching at both population and finite-sample levels. We further specialize the framework to group-symmetric targets, providing the first error analysis of equivariant flow-based models and the first quantitative comparison between data augmentation and equivariant inductive bias. Our analysis identifies a symmetry-aware Wasserstein path-space divergence that quantifies the model-form error induced by non-equivariant parametrizations. We prove that this error cannot be removed by additional data or training and vanishes only under equivariant architectures, establishing a precise theoretical advantage of equivariant inductive bias over data augmentation. Numerical experiments on group-symmetric Gaussian mixtures corroborate the theory.