flow matching

Existing mean-field approaches enable single-step generation for accelerated inference but struggle to accurately replicate the multi-step flow matching dynamics in continuous data modeling, leading to degraded generation fidelity and diversity. To address this, we propose OT-MeanFlow, an enhanced mean-field framework grounded in optimal transport (OT) theory. By designing theoretically principled transport paths, OT-MeanFlow ensures that single-step generation faithfully approximates the underlying multi-step flow matching dynamics—without increasing sampling steps. This preserves computational efficiency while substantially improving sample quality. Experiments across image generation, image-to-image translation, and point cloud generation demonstrate that OT-MeanFlow consistently outperforms state-of-the-art mean-field and accelerated sampling baselines under single-step evaluation. These results validate the effectiveness and broad applicability of OT-guided simplification for flow matching.

This work addresses the inefficiency of traditional diffusion and flow-matching models, which require costly iterative inference, and the tendency of direct transport map learning to suffer from mean collapse. The authors propose OT-NFM, a framework that parameterizes the optimal transport–induced flow map directly via neural flows, enabling single-step generation without solving ordinary differential equations. By introducing scalable minibatch and online optimal transport coupling strategies, the method enforces consistent pairing between noise and data samples within an end-to-end differentiable training pipeline, thereby avoiding degenerate solutions. Empirical results on MNIST, CIFAR-10, and synthetic benchmarks demonstrate that the model achieves competitive generation quality with only a single forward pass.

Generative models such as diffusion models rely on multi-step numerical integration for inference, incurring high computational cost; while consistency models enable one-step generation, they lack a unified theoretical foundation. Method: We propose a novel paradigm that directly learns the flow map between two time points of an ordinary differential equation (ODE), unifying consistency modeling, progressive distillation, and other few-step generation approaches. Leveraging stochastic interpolation, we jointly optimize flow map prediction and velocity field distillation loss, integrating ODE theory with neural operator principles to enable tunable step counts for precision–efficiency trade-offs. Contribution/Results: On CIFAR-10 and ImageNet 32×32, our method achieves 10–50× speedup in sampling over standard diffusion models while preserving competitive sample quality—demonstrating both theoretical coherence and practical efficacy in accelerating generative inference.

Contemporary continuous-time generative models typically exhibit a “V-shaped” transport: samples evolve independently from the prior to data points via straight-line trajectories, ignoring intrinsic shared structures among data. To address this, we propose Y-Flow—a Y-shaped generative flow that explicitly models hierarchical data relationships through a shared latent path: mass first moves collectively along a common trajectory, then bifurcates toward individual targets. Our key innovations include (i) a sublinear exponential velocity field that reduces transport cost and promotes efficient collective mass transport, breaking the V-shaped paradigm; and (ii) a scalable flow-matching objective built upon neural ordinary differential equations. Experiments on synthetic, image, and single-cell biological datasets demonstrate that Y-Flow significantly outperforms mainstream baselines—achieving superior distributional fidelity (e.g., lower FID and MMD) and improved sampling efficiency (reducing ODE integration steps by over 50%).

This work addresses the modeling and learning of velocity fields governing data distribution evolution in flow matching, aiming to unify transport planning, Markov kernels, and stochastic process paradigms. Methodologically, it establishes the first theoretical equivalence framework for velocity fields characterizing absolutely continuous Wasserstein curves across these three constructions; introduces the conditional Wasserstein distance as a novel metric for Bayesian inverse problems; and unifies the geometric interpretations of continuous normalizing flows and score matching. Leveraging tools from Wasserstein geometry, optimal transport, and stochastic differential equations, the paper rigorously proves the intrinsic consistency of multiple velocity field learning approaches, thereby strengthening the mathematical foundations of flow matching. Experiments demonstrate that the proposed framework effectively generates high-fidelity conditional distributions in Bayesian inverse problems.

This work addresses the challenge of achieving high-quality generation with drastically reduced inference steps, enabling efficient one-step or few-step synthesis. The authors propose a unified few-step generation framework based on cumulative flow matching, which introduces cumulative flow abstraction and cumulative parameterization to jointly model local instantaneous updates and global probability transport over finite time horizons. Notably, this approach requires no increase in model capacity or architectural modifications, making it readily applicable to a broad range of diffusion and flow-based models. Empirical evaluations across diverse tasks—including image generation, geometric distribution modeling, joint prediction, and signed distance field (SDF) synthesis—demonstrate that the framework achieves compelling generation quality at substantially lower inference costs, highlighting its strong versatility and computational efficiency.

This work addresses the limitations of conventional diffusion models in stochastic closure modeling—namely, slow sampling and the trade-off between physical fidelity and data efficiency—by introducing a transport-based generative approach that enables single-step fast sampling through flow matching on a low-dimensional latent manifold. By combining explicit geometric regularization (e.g., metric-preserving or geometry-aware constraints) with implicit regularization via joint training, the method effectively controls latent space deformation to preserve physical consistency. With only limited training data, it accurately reconstructs the topological structure and physical properties of the underlying dynamical system, achieving sampling speeds two orders of magnitude faster than iterative diffusion models.

While existing single-step diffusion models are efficient, they lack the flexibility to balance generation quality and computational cost. To address this limitation, this work proposes the Drift Flow Matching framework, which unifies single-step drift modeling with multi-step flow matching for the first time. By integrating the efficiency of direct transport maps with the iterative refinement capability of continuous flows, the framework establishes a new generative paradigm that enables on-demand trade-offs between computational expenditure and output fidelity. Experimental results across diverse tasks and datasets demonstrate that the proposed method achieves a flexible and effective balance between efficiency and quality, exhibiting both strong generalizability and high performance.

This work addresses the problem of constructing an exact one-step generative mapping between a source distribution and a singular target distribution supported on a low-dimensional manifold. To this end, it proposes a flow matching approach based on time-independent (autonomous) velocity fields, which directly learns the desired mapping through a conservation equation and provides a dynamical interpretation of the flux constraint arising in the Beckmann formulation of optimal transport. Theoretically, the paper establishes the equivalence between autonomous flows and one-step mappings, unifying Poisson flow generative models with equilibrium matching under quadratic regression loss, thereby resolving inconsistencies in existing methods. Empirically, the proposed approach demonstrates its effectiveness through high-quality image generation on ImageNet at 256×256 resolution.

This work systematically investigates the impact of loss weighting strategies and output parameterizations on model performance in flow matching. Through numerical experiments on both synthetic data with controllable geometric structures and real-world images, the study disentangles their interaction effects across varying data manifold dimensions, model architectures, and dataset scales, using PSNR and FID as evaluation metrics. The analysis reveals, for the first time, how the optimal choice of loss weighting and parameterization depends critically on the intrinsic structure of the data. Building on these insights, the authors formulate practical design principles that substantially improve denoising accuracy and generation quality.