Flow-Based Generative Modeling

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Overview

Flow-based generative modeling represents generation as a transformation of probability mass, often by integrating a continuous-time velocity field from noise to data or composing invertible normalizing-flow layers. Flow matching and rectified flow train velocity fields by regression along selected paths; optimal-transport couplings, stochastic interpolants, discrete Markov flows, and flow maps extend the toolkit to different data types and sampling budgets. Researchers apply these methods to images, language, molecules, physical fields, and robot actions, assessing fidelity and diversity alongside likelihood, constraint satisfaction, calibration, and runtime. A central challenge is achieving one- or few-step generation without sacrificing quality or mode coverage, while improving path design, robustness, geometric and physical consistency, and theoretical guarantees.

Latest papers

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563 papers125 papers in the last four weeks, up 131% on the four weeks before. 0.7% of all new papers.

May 9, 2026

Deterministic Decomposition of Stochastic Generative Dynamics

Modern generative models can be understood as probability transport from a simple base distribution to a target data distribution. Deterministic transport models offer tractable velocity-field parameterizations, whereas stochastic generative models capture richer density evolution through drift and diffusion. Yet when stochastic dynamics are described through deterministic velocity fields, the effects of drift and diffusion are often compressed into a single effective field, obscuring the distinct roles of deterministic evolution and stochastic fluctuation. In this work, we show that the deterministic field btb_t of a stochastic generative process admits a natural transport--osmotic decomposition that separates deterministic transport from stochastic, diffusion-induced effects: bt=ut+dtb_t = u_t + d_t, where utu_t governs marginal probability transport and dtd_t captures an osmotic effect induced by diffusion and determined by the marginal score. Based on this decomposition, we propose Bridge Matching, a flow-based framework for learning decomposed generative dynamics through both marginal and conditional formulations. In generative modeling experiments, we recombine the learned components as bt=ut+λddtb_t = u_t + λ_d d_t, showing that the proposed decomposition enables interpretable and controllable sampling by adjusting the osmotic contribution in probability transport.

Xingyu Song, Yuan Mei, Naoya Takeishi

1The University of Tokyo · 2Zhejiang University · *Visiting student at The University of Tokyo from Zhejiang University.

Flow MatchingStochastic Differential EquationsGenerative ModelingFlow-Based Generative Modeling
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May 8, 2026

Generalized Wasserstein Flow Matching: Nested Couplings for Point Cloud Generation

We formulate flow matching on probability measures over probability measures. Directly from the Wasserstein-on-Wasserstein (WoW) continuity equation, we show that admissible nested outer and inner couplings induce a valid metameasure flow and non-local velocity field. Validity requires neither optimal plans nor a smooth manifold. Practically, this opens up the design space of multisample flow matching: exact Wasserstein plans are distinguished by energy and straightness, while sliced and lazy linear plans yield scalable alternatives. Point-cloud experiments, including controlled Bures-Wasserstein and Euclidean minibatch baselines, demonstrate the resulting geometry-cost trade-off.

Moritz Piening, Richard Duong, Gabriele Steidl

Institut für Mathematik, Technische Universität Berlin, Germany

Flow MatchingGenerative ModelingPoint Cloud GenerationFlow-Based Generative ModelingOptimal Transport
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May 8, 2026

Exploring and Exploiting Stability in Latent Flow Matching

In this work, we show that Latent Flow-Matching (LFM) models are robust to different types of perturbations, including data reduction and model capacity shrinkage. We characterize this stability by these models' tendency to generate similar outputs under identical noise seeds. We provide a perspective relating this phenomenon to flow matching theory, which indicates that this stability is inherent to the FM objective. We further exploit this stability to derive practical algorithms for more efficient training and inference. Concretely, first, we show that by training LFM models on significantly reduced datasets, performance is preserved, and in compute-constrained regimes, the model converges faster while maintaining quality. This yields multiple advantages, including savings in the training time due to faster convergence, and alleviating annotation effort when training conditional models. Second, LFM stability under architectural shrinkage gives rise to a two-model coarse-to-fine approach, one using a light-weight architecture for the first phase of the FM trajectory, and one with higher capacity for the second, thereby reducing the inference cost substantially. To determine which samples are informative, we introduce three sample-scoring criteria and evaluate them under standard metrics for generative models. Our results are thoroughly evaluated on multiple datasets, demonstrating the practical advantage of this stability, including data savings and a more than two-fold inference speedup while generating comparable outputs.

Rania Briq, Michael Kamp, Ohad Fried +2

Forschungszentrum Jülich · Technical University Dortmund · Lamarr Institute +4

Flow MatchingEfficient InferenceGenerative ModelingFlow-Based Generative Modeling
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May 8, 2026

What Time Is It? How Data Geometry Makes Time Conditioning Optional for Flow Matching

Recent work has shown that models flow matching models can be trained without explicit time conditioning, challenging the standard view that the interpolation time is needed to disambiguate velocity targets. But why should a time-blind model work at all? Decomposing the time-blind flow matching loss, we identify two sources of irreducible error: a coupling variance, which arises from ambiguous velocity targets induced by how noise and data points are paired, and the time-blindness gap, which is the additional error caused by ignoring time. This gap shows that time-blind training is strictly harder than conventional training, reinforcing the puzzle that time-blind models work so well in practice. We resolve this tension by showing that the geometry of high-dimensional data makes time identifiable directly from noisy observations. When data concentrates near a kk-dimensional subspace, time can be recovered from the statistical structure of noisy interpolants in directions orthogonal to the data; under a spiked-covariance model, this yields a closed-form estimator that recovers tt from a single observation zz at rate O(1/d−k)O(1/\sqrt{d-k}) for ambient dimension dd. As a consequence, we prove that the time-blindness gap is asymptotically negligible relative to the coupling variance. We empirically demonstrate our identifiability result on real-world data and show that changing the coupling has a much larger effect on loss and sample quality than removing time conditioning across CIFAR-10, CelebA-HQ, and FFHQ. These results explain why time-blind flow matching works and show that the main practical lever is the choice of coupling, not explicit time conditioning.

Alec Helbling, Sebastian Gutierrez Hernandez, Benjamin Hoover +2

Georgia Tech · IBM Research

Flow MatchingFlow-Based Generative Modeling
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May 8, 2026

Normalizing Trajectory Models

Diffusion-based models decompose sampling into many small Gaussian denoising steps -- an assumption that breaks down when generation is compressed to a few coarse transitions. Existing few-step methods address this through distillation, consistency training, or adversarial objectives, but sacrifice the likelihood framework in the process. We introduce Normalizing Trajectory Models (NTM), which models each reverse step as an expressive conditional normalizing flow with exact likelihood training. Architecturally, NTM combines shallow invertible blocks within each step with a deep parallel predictor across the trajectory, forming an end-to-end network trainable from scratch or initializable from pretrained flow-matching models. Its exact trajectory likelihood further enables self-distillation: a lightweight denoiser trained on the model's own score produces high-quality samples in four steps. On text-to-image benchmarks, NTM matches or outperforms strong image generation baselines in just four sampling steps while uniquely retaining exact likelihood over the generative trajectory.

Jiatao Gu, Tianrong Chen, Ying Shen +3

Apple · UIUC

Normalizing FlowsDiffusion Model DistillationFew-Step Diffusion SamplingDiffusion-Based Image GenerationFlow-Based Generative ModelingDiffusion Models
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May 8, 2026

Flow-OPD: On-Policy Distillation for Flow Matching Models

Existing Flow Matching (FM) text-to-image models suffer from two critical bottlenecks under multi-task alignment: the reward sparsity induced by scalar-valued rewards, and the gradient interference arising from jointly optimizing heterogeneous objectives, which together give rise to a 'seesaw effect' of competing metrics and pervasive reward hacking. Inspired by the success of On-Policy Distillation (OPD) in the large language model community, we propose Flow-OPD, the first unified post-training framework that integrates on-policy distillation into Flow Matching models. Flow-OPD adopts a two-stage alignment strategy: it first cultivates domain-specialized teacher models via single-reward GRPO fine-tuning, allowing each expert to reach its performance ceiling in isolation; it then establishes a robust initial policy through a Flow-based Cold-Start scheme and seamlessly consolidates heterogeneous expertise into a single student via a three-step orchestration of on-policy sampling, task-routing labeling, and dense trajectory-level supervision. We further introduce Manifold Anchor Regularization (MAR), which leverages a task-agnostic teacher to provide full-data supervision that anchors generation to a high-quality manifold, effectively mitigating the aesthetic degradation commonly observed in purely RL-driven alignment. Built upon Stable Diffusion 3.5 Medium, Flow-OPD raises the GenEval score from 63 to 92 and the OCR accuracy from 59 to 94, yielding an overall improvement of roughly 10 points over vanilla GRPO, while preserving image fidelity and human-preference alignment and exhibiting an emergent 'teacher-surpassing' effect. These results establish Flow-OPD as a scalable alignment paradigm for building generalist text-to-image models. The codes and weights will be released in: https://github.com/CostaliyA/Flow-OPD .

Zhen Fang, Wenxuan Huang, Yu Zeng +8

University of Science and Technology of China · University of California, Los Angeles · 3The Chinese University of Hong Kong +1

Diffusion Model AlignmentFlow MatchingRL for Image GenerationFlow-Based Generative ModelingOn-Policy DistillationT2I Diffusion Models
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May 8, 2026

STARFlow2: Bridging Language Models and Normalizing Flows for Unified Multimodal Generation

Deep generative models have advanced rapidly across text and vision, motivating unified multimodal systems that can understand, reason over, and generate interleaved text-image sequences. Most existing approaches combine autoregressive language modeling with diffusion-based image generators, inheriting a structural mismatch between causal text generation and iterative visual denoising. We observe that autoregressive normalizing flows are autoregressive Transformers--sharing the same causal mask, KV-cache mechanism, and left-to-right structure as LLMs--making them the most natural paradigm for true unified multimodal generation. We present STARFlow2, built on the Pretzel architecture that vertically interleaves a pretrained VLM stream with a TarFlow stream via residual skip connections, both operating under the same causal mask. Combined with a deep-shallow flow design and a unified FAE latent space, STARFlow2 enables cache-friendly interleaved generation where both text and visual outputs directly enter the KV-cache without re-encoding. Experiments demonstrate strong performance across image generation and multimodal understanding benchmarks, validating autoregressive flows as a viable foundation for unified multimodal modeling.

Ying Shen, Tianrong Chen, Yuan Gao +6

UIUC · Apple

Unified Multimodal ModelsMultimodal GenerationAutoregressive Image GenerationFlow-Based Generative ModelingMultimodal Foundation Models
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May 8, 2026

Trajectory as the Teacher: Few-Step Discrete Flow Matching via Energy-Navigated Distillation

Discrete flow matching generates text by iteratively transforming noise tokens into coherent language, but may require hundreds of forward passes. Distillation uses the multi-step trajectory to train a student to reproduce the process in a few steps. When the student underperforms, the usual explanation is insufficient capacity. We argue the opposite: the trajectory is the bottleneck, not the student. Each training trajectory is built through a chain of blind stochastic jumps with no evaluation of sequence quality; a single bad decision at an early midpoint propagates through subsequent steps, yet the student must imitate the result. Trajectory-Shaped Discrete Flow Matching (TS-DFM) replaces these blind jumps with guided navigation: a lightweight energy compass evaluates candidate continuations at each midpoint, selecting the most coherent. All shaping is training-only; inference cost is unchanged. On 170M-parameter language modeling, the shaped student at 8 steps achieves 32% lower perplexity than the 1,024-step teacher while being 128x faster, with gains consistent across source distributions and three evaluators of increasing scale. TS-DFM achieves the best perplexity of any discrete-generation baseline we compare against, including methods trained on 6x more data or using 5x larger models.

Amin Karimi Monsefi, Dominic Culver, Nikhil Bhendawade +3

1The Ohio State University · 2Apple

Flow MatchingDiffusion Language Model InferenceLanguage Model DistillationFlow-Based Generative ModelingTeacher-Student LearningDiscrete Flow Matching
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May 8, 2026

Consistency Regularised Gradient Flows for Inverse Problems

Vision-Language Latent Diffusion Models (LDMs) (Rombach et al., 2022) provide powerful generative priors for inverse problems. However, existing LDM-based inverse solvers typically require a large number of neural function evaluations (NFEs) and backpropagation through large pretrained components, leading to substantial computational costs and, in some cases, degraded reconstruction quality. We propose a unified Euclidean-Wasserstein-2 gradient-flow framework that jointly performs posterior sampling and prompt optimization in the latent space through a single flow that aligns the prior and posterior with the observed data. Combined with few-step latent text-to-image models, this formulation enables low-NFE inference without backpropagation through autoencoders. Experiments across several canonical imaging inverse problems show that our method achieves state-of-the-art performance with significantly reduced computational cost.

Alessio Spagnoletti, Tim Y. J. Wang, Marcelo Pereyra +1

Laboratoire MAP5, UMR 8145, Université Paris Cité, CNRS · Department of Mathematics, Imperial College London · Heriot-Watt University, MACS & Maxwell Institute

Latent Diffusion ModelsWasserstein Gradient FlowsDiffusion-Based Inverse ProblemsFlow-Based Generative ModelingDiffusion Posterior SamplingImage Inverse Problems
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May 8, 2026

Scaling Categorical Flow Maps

Continuous diffusion and flow matching models could represent a powerful alternative to autoregressive approaches for language modelling (LM), as they unlock a host of advantages currently reserved for continuous modalities, including accelerated sampling and tilting. Recently, several works have demonstrated the possibility of generating discrete data continuously by a simple flow matching process between a Gaussian and the one-hot encoded data distribution. They have further shown the feasibility of accelerated sampling via Categorical Flow Maps (CFMs), resulting in competitive sample quality in the few-step regime. However, this method had only been evaluated at relatively modest scales (<1<1B), leaving the question of its scalability completely open. In this article, we train a 1.71.7B-parameter base flow model on 2.12.1T tokens and self-distill it into a CFM that generates diverse, high-quality text in as few as 44 inference steps while maintaining near-data-level token entropy. Furthermore, we introduce a likelihood bound for CFMs in the semi-discrete setting, and show that they can be used to score the model on standard LM benchmarks, achieving results in the same range as discrete diffusion methods. Finally, we uncover some of the challenges that arise from training these models at scale, and we provide prescriptive insights on loss weighting and time scheduling.

Oscar Davis, Anastasiia Filippova, Pierre Ablin +4

Apple · University of Oxford

Language Model ScalingFlow Map LearningFlow-Based Generative ModelingContinuous Diffusion Language ModelsText GenerationDiscrete Flow Matching
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May 8, 2026

Flow Matching for Count Data

High-dimensional count data arise in applications such as single-cell RNA sequencing and neural spike trains, where mappings between distributions across successive batches or time points form critical components of data analysis. The recent success of diffusion- and flow-based deep generative models for images, video, and text motivates extending these ideas to count-valued settings, but many existing methods either treat each count as a categorical state or transform counts into a continuous space, neither of which is natural or efficient when the count range is large. We propose count-FM, a flow-matching framework for count data based on a continuous-time birth-death process with local unit jumps. Count-FM learns marginal transitions efficiently in count space through simulation-free training of conditional transition rates, allowing transport between arbitrary count-distributed source and target populations. In simulation, count-FM variants achieve strong sample quality while using substantially fewer parameters. We further apply count-FM to scRNA-seq and neural spike-train data for unconditional generation, transport, and conditional generation. Across these tasks, count-FM yields improved sample quality, greater modeling efficiency, and interpretable transport paths.

Ganchao Wei, John Pearson

Department of Neurobiology Department of Statistical Science Duke University Durham, NC, USA · Department of Neurobiology Department of Electrical and Computer Engineering Duke University Durham, NC, USA

Flow MatchingSynthetic Data GenerationFlow-Based Generative Modeling
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May 8, 2026

Structured Coupling for Flow Matching

Standard flow matching scales well but typically relies on an unstructured source distribution, limiting its ability to learn interpretable latent structure. Latent-variable models, by contrast, capture structure but often sacrifice generative quality. We bridge this gap by proposing Structured Coupling for Flow Matching (SCFM), a cooperative framework that augments flow matching with structured latent representation learning. By introducing structured latent variables and exogenous noise into the source, SCFM jointly learns a structured prior (via latent variable modeling) and a continuous transport map (via flow matching). It uses a shared time-dependent recognition network for both latent variable model variational inference and intermediate-time flow velocity estimation. This yields a structurally informed yet unconditional, simulation-free flow model, where the latent variable model can also assist flow sampling. Empirically, SCFM facilitates unsupervised latent representation learning for clustering, disentanglement and downstream tasks, while remaining competitive with flow matching in sample quality, showing that meaningful structure can be learned without sacrificing generative fidelity.

Xavier Sumba, Carles Balsells-Rodas, Yingzhen Li

Imperial College London

Flow MatchingDisentangled Representation LearningLatent Variable ModelsUnsupervised Representation LearningGenerative ModelingFlow-Based Generative Modeling
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May 8, 2026

Flow-Direct: Feedback-Efficient and Reusable Guidance for Flow Models via Non-Parametric Guidance Field

Training-free guidance enables pre-trained diffusion and flow models to optimize application-specific objectives using feedback from external black-box reward functions. However, existing methods are feedback-inefficient because reward feedback is used only transiently to inform a localized gradient approximation or a discrete search decision, and is subsequently discarded. To address this limitation, we propose Flow-Direct, a framework that guides the generation process via a persistent guidance field. Theoretically, this guidance field is analytically derived from the log-density ratio between the base and reward-weighted target distributions; it transports the pre-trained distribution to the target distribution. In practice, the field is implemented as a non-parametric estimator constructed from all accumulated reward-evaluated samples. As more samples are collected during optimization, this empirical guidance field becomes increasingly accurate. This persistent formulation yields two major advantages. First, Flow-Direct is highly feedback-efficient: because every evaluated sample is used to refine the global guidance field, no reward information is wasted. Second, the framework is naturally reusable: once optimization is complete, the collected dataset defines a reusable guidance field for generating novel target samples without additional reward evaluations, and distinct guidance fields can be combined to generate samples that simultaneously satisfy multiple objectives.

Kim Yong Tan, Yueming Lyu, Ivor Tsang +1

College of Computing and Data Science, Nanyang Technological University, Singapore · Centre for Frontier AI Research, Agency for Science, Technology and Research, Singapore · Institute of High Performance Computing, Agency for Science, Technology and Research, Singapore

Diffusion Model GuidanceBlack-Box OptimizationFlow-Based Generative ModelingInference-Time Guidance
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May 7, 2026

Conservative Flows: A New Paradigm of Generative Models

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.

Eshed Gal, Md Shahriar Rahim Siddiqui, Moshe Eliasof +1

The University of British Columbia Vancouver, Canada · University of Cambridge Cambridge, United Kingdom

Langevin DynamicsGenerative Flow NetworksGenerative ModelingFlow-Based Generative Modeling
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May 7, 2026

A Unified Measure-Theoretic View of Diffusion, Score-Based, and Flow Matching Generative Models

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 (ρt)t∈[0,1](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.

Aditya Ranganath, Mukesh Singhal

Center for Applied Scientific Computing Lawrence Livermore National Laboratory Livermore, CA 94551, USA · Department of Electrical Engineering University of California Merced, CA 95343, USA

Flow MatchingGenerative ModelingFlow-Based Generative ModelingScore-Based Generative ModelingDiffusion Models
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May 7, 2026

Improved techniques for fine-tuning flow models via adjoint matching: a deterministic control pipeline

We propose a deterministic adjoint matching framework that formulates human preference alignment for flow-based generative models as an optimal control problem over velocity fields. One can directly regress the control toward a value-gradient-induced target under the current policy, leading to a simple and stable training objective. Building on this perspective, we introduce a truncated adjoint scheme that focuses computation on the terminal portion of the trajectory, where reward-relevant signals concentrate, which yields substantial computational savings while preserving alignment quality. We further generalize the framework beyond standard KL-based regularization, allowing more flexible trade-offs between alignment strength and distributional preservation. Experiments on SiT-XL/2 and FLUX.2-Klein-4B demonstrate consistent gains across multiple alignment metrics, along with substantially improved diversity and mode preservation.

Zhengyi Guo, Jiayuan Sheng, David D. Yao +1

Fine-TuningFlow-Based Generative Modeling
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May 7, 2026

FREPix: Frequency-Heterogeneous Flow Matching for Pixel-Space Image Generation

Pixel-space diffusion has re-emerged as a promising alternative to latent-space generation because it avoids the representation bottleneck introduced by VAEs. Yet most existing methods still treat image generation as a frequency-homogeneous process, overlooking the distinct roles and learning dynamics of low- and high-frequency components. To address this, we propose FREPix, a FREquency-heterogeneous flow matching framework for Pixel-space image generation. FREPix explicitly decomposes generation into low- and high-frequency components, assigns them separate transport paths, predicts them with a factorized network, and trains them with a frequency-aware objective. In this way, coarse-to-fine generation becomes an explicit design principle rather than an implicit behavior. On ImageNet class-to-image generation, FREPix achieves competitive results among pixel-space generation models, reaching 1.91 FID at 256×256256\times256 and 2.38 FID at 512×512512\times512, with particularly strong behavior in the low-NFE regime.

Mingfeng Lin, Jiakun Chen, Liang Han +1

Harbin Institute of Technology (Shenzhen)

Flow MatchingConditional Image GenerationImage GenerationPixel-Space Diffusion ModelsFlow-Based Generative Modeling
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May 7, 2026

Flow Matching with Arbitrary Auxiliary Paths

We introduce a new generative modeling framework, Flow Matching with Arbitrary Auxiliary Paths (AuxPath-FM), which generalizes conditional flow matching by incorporating an auxiliary variable drawn from an arbitrary distribution into the probability path. Unlike prior methods that restrict auxiliary components to Gaussian noise, AuxPath-FM allows the variable ηη to follow any distribution, producing trajectories of the form Xt=a(t)X1+b(t)X0+c(t)ηX_t = a(t)X_1 + b(t)X_0 + c(t)η. We theoretically demonstrate that this construction preserves the continuity equation and maintains a training objective consistent with the marginal formulation. This flexibility enables the design of diverse probability paths using various priors, including Gaussian, Uniform, Laplace, and discrete Rademacher distributions, each offering unique geometric properties for generative flows. Furthermore, our framework allows for specialized tasks such as label-guided generation by encoding structured semantic information into the auxiliary distribution. Overall, AuxPath-FM provides a principled and general foundation for probability path design, offering both theoretical generality and practical flexibility for diverse generative modeling tasks.

Xin Peng, Ang Gao

School of Physical Science and Technology, Beijing University of Posts and Telecommunications, Beijing, China

Conditional Flow MatchingFlow-Based Generative ModelingConditional Generative Modeling
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May 7, 2026

A Flow Matching Algorithm for Many-Shot Adaptation to Unseen Distributions

While generative modeling has achieved remarkable success on tasks like natural language-conditioned image generation, enabling model adaptation from example data points remains a relatively underexplored and challenging problem. To this end, we propose Function Projection for Flow Matching (FP-FM), an algorithm that directly conditions generation on samples from the target distribution. FP-FM learns basis functions to span the velocity fields corresponding to a set of training distributions, and adapts to new distributions by computing a simple least-squares projection onto this basis. This enables efficient generation of samples from diverse target distributions without additional training at inference time. We further introduce multiple variants of FP-FM that provide a trade-off in expressivity and compute by enriching the coefficient calculation, e.g., by making the coefficients dependent on time. FP-FM achieves greatly improved precision and recall relative to baselines across synthetic and image-based datasets, with especially strong gains on unseen distributions.

Tyler Ingebrand, Ruihan Zhao, Kushagra Gupta +3

University of Texas at Austin

Flow MatchingFlow-Based Generative ModelingConditional Generative Modeling
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May 7, 2026

P-Guide: Parameter-Efficient Prior Steering for Single-Pass CFG Inference

Classifier-Free Guidance (CFG) is essential for high-fidelity conditional generation in flow matching, yet it imposes significant computational overhead by requiring dual forward passes at each sampling step. In this work, we address this bottleneck by introducing P-Guide, a framework that achieves high-quality guidance through a single inference pass by modulating only the initial latent state. We further show that, under a first-order approximation, P-Guide is equivalent to CFG in the sense that it steers generation from the prior space, without requiring explicit velocity field extrapolation during sampling. We consider both homoscedastic and heteroscedastic priors, and find that jointly modeling the mean and variance enables adaptive loss attenuation and improved robustness to data uncertainty. Extensive experiments demonstrate that P-Guide reduces inference latency by approximately 50% while maintaining fidelity and prompt alignment competitive with standard dual-pass CFG baselines.

Xin Peng, Ang Gao

School of Physical Science and Technology, Beijing University of Posts and Telecommunications, Beijing, China

Flow MatchingClassifier-Free GuidanceFlow-Based Generative ModelingDiffusion Model Inference Acceleration
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May 7, 2026

Physical Fidelity Reconstruction via Improved Consistency-Distilled Flow Matching for Dynamical Systems

Reconstructing high-fidelity flow fields from low-fidelity observations is a central problem in scientific machine learning, yet recent diffusion and flow-matching models typically rely on iterative sampling, making them costly for latency-sensitive workflows such as ensemble forecasting, real-time visualization, and simulation-in-the-loop inference. We study whether a high-fidelity flow-matching generative model can be compressed into a compact one-step model for fast scientific flow reconstruction. Our approach distills an optimal-transport flow-matching teacher into a one-step consistency model. Low-fidelity observations are incorporated at inference by initializing the generative trajectory from a noised observation along the transport path, allowing an unconditional high-fidelity flow model to perform conditional reconstruction without retraining the teacher. We evaluate this distillation strategy on three fluid benchmarks, Smoke Buoyancy, Turbulent Channel Flow, and Kolmogorov Flow, using coarse-to-fine reconstruction as a controlled testbed at field sizes up to 256×256256 \times 256. Across these settings, the distilled student retains similar performance of the teacher's model on spectrum metrics, while using roughly half as many parameters and achieving a 12×12\times inference speedup over the flow-matching teacher. Under the same training budget, the distilled student also outperforms a one-step consistency model trained directly from scratch by 23.1%23.1\% in SSIM, showing that teacher distillation improves training efficiency rather than merely accelerating sampling. These results suggest a promising route for turning future high-capacity scientific generative models into compact reconstruction models that are faster to train, cheaper to run, and easier to deploy.

Sicheng Ma, Tianyue Yang, Xiuzhe Wu +1

Department of Physics University of Cambridge · The Center for Computational Science University College London · Department of Computer Science University College London

Flow MatchingOne-Step Generative ModelingConsistency DistillationFlow-Based Generative Modeling
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May 7, 2026

SDFlow: Similarity-Driven Flow Matching for Time Series Generation

Vector quantization (VQ) with autoregressive (AR) token modeling is a widely adopted and highly competitive paradigm for time-series generation. However, such models are fundamentally limited by exposure bias: during inference, errors can accumulate across sequential predictions, leading to pronounced quality degradation in long-horizon generation. To address this, we propose SDFlow (S\textbf{S}imilarity-D\textbf{D}riven Flow\textbf{Flow} Matching), a non-autoregressive framework that operates entirely in the frozen VQ latent space and enables parallel sequence generation via flow matching. We tackle three key challenges in making this transition: (1) eliminating exposure bias by replacing step-wise token prediction with a global transport map; (2) mitigating the high-dimensionality of VQ token spaces via a low-rank manifold decomposition with a learned anchor prior over the latent manifold; and (3) incorporating discrete supervision into continuous transport dynamics by introducing a categorical posterior over codebook indices within a variational flow-matching formulation. Extensive experiments show that SDFlow achieves state-of-the-art performance, improving Discriminative Score and substantially reducing Context-FID, particularly for challenging long-sequence generation. Moreover, SDFlow provides significant inference speedups over autoregressive baselines, offering both high fidelity and computational efficiency. Code is available at https://github.com/William-Liwei/SDFlow

Wei Li, Shibo Feng, Pengcheng Wu +3

Shanghai Jiao Tong University · Shanghai University · Nanyang Technological University +3

Flow MatchingTime Series GenerationFlow-Based Generative Modeling
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May 7, 2026

Spherical Flows for Sampling Categorical Data

We study the problem of learning generative models for discrete sequences in a continuous embedding space. Whereas prior approaches typically operate in Euclidean space or on the probability simplex, we instead work on the sphere Sd−1\mathbb S^{d-1}. There the von Mises-Fisher (vMF) distribution induces a natural noise process and admits a closed-form conditional score. The conditional velocity is in general intractable. Exploiting the radial symmetry of the vMF density we reduce the continuity equation on Sd−1\mathbb S^{d-1} to a scalar ODE in the cosine similarity, whose unique bounded solution determines the velocity. The marginal velocity and marginal score on (Sd−1)L(\mathbb S^{d-1})^L both decompose into posterior-weighted tangent sums that differ only by per-token scalar weights. This gives access to both ODE and predictor-corrector (PC) sampling. The posterior is the only learned object, trained by a cross-entropy loss. Experiments compare the vMF path against geodesic and Euclidean alternatives. The combination of vMF and PC sampling significantly improves results on Sudoku and language modeling.

Jannis Chemseddine, Gregor Kornhardt, Gabriele Steidl

Technische Universität Berlin

Flow MatchingSynthetic Data GenerationGenerative ModelingFlow-Based Generative ModelingSequence Modeling
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May 6, 2026

HDFlow: Hierarchical Diffusion-Flow Planning for Long-horizon Tasks

Recent advances in generative models have shown promise in generating behavior plans for long-horizon, sparse reward tasks. While these approaches have achieved promising results, they often lack a principled framework for hierarchical decomposition and struggle with the computational demands of real-time execution, due to their iterative denoising process. In this work, we introduce Hierarchical Diffusion-Flow (HDFlow), a novel hierarchical planning framework that optimally leverages the strengths of diffusion and rectified flow models to overcome the limitations of single-paradigm generative planners. HDFlow employs a high-level diffusion planner to generate sequences of strategic subgoals in a learned latent space, capitalizing on diffusion's powerful exploratory capabilities. These subgoals then guide a low-level rectified flow planner that generates smooth and dense trajectories, exploiting the speed and efficiency of ordinary differential equation (ODE)-based trajectory generation. We evaluate HDFlow on four challenging furniture assembly tasks in both simulation and real-world, where it significantly outperforms state-of-the-art methods. Furthermore, we also showcase our method's generalizability on two long-horizon benchmarks comprising diverse locomotion and manipulation tasks. Project website: https://hdflow-page.github.io/

Nandiraju Gireesh, Yuanliang Ju, Chaoyi Xu +3

1Peking University · University of Toronto. · 2Galbot

Hierarchical PlanningLong-Horizon Robotic ManipulationTrajectory GenerationDiffusion-Based PlanningRobot Motion GenerationFlow-Based Generative Modeling
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May 5, 2026

A Few-Step Generative Model on Cumulative Flow Maps

We propose a unified, few-step generative modeling framework based on cumulative flow maps for long-range transport in probability space, inspired by flow-map techniques for physical transport and dynamics. At its core is a cumulative-flow abstraction that connects local, instantaneous updates with finite-time transport, enabling generative models to reason about global state transitions. This perspective yields a unified few-step framework built on cumulative transport and \revise{cumulative} parameterization that applies broadly to existing diffusion- and flow-based models without being tied to a specific prediction \revise{instantiation}. Our formulation supports few-step and even one-step generation while preserving synthesis quality, requiring only minimal changes to time embeddings and training objectives, and no increase in model capacity. We demonstrate its effectiveness across diverse tasks, including image generation, geometric distribution modeling, joint prediction, and SDF generation, with reduced inference cost.

Zhiqi Li, Duowen Chen, Yuchen Sun +1

Georgia Institute of Technology, Atlanta, United States of America

One-Step Generative ModelingFlow-Based Generative ModelingDiffusion Model Acceleration
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May 5, 2026

PerFlow: Physics-Embedded Rectified Flow for Efficient Reconstruction and Uncertainty Quantification of Spatiotemporal Dynamics

Reconstructing PDE-governed fields from sparse and irregular measurements is challenging due to their ill-posed nature. Deterministic surrogates are trained on dense fields that struggle with limited measurements and uncertainty quantification. Generative models, by learning distributions over spatiotemporal fields, can better handle sparsity and uncertainty. However, existing generative approaches enforce data consistency and PDE constraints simultaneously via sampling-time gradient guidance, resulting in slow and unstable inference. To this end, we propose PerFlow, a Physics-embedded rectified Flow for efficient sparse reconstruction and uncertainty quantification of spatiotemporal dynamics. PerFlow decouples observation conditioning from physics enforcement, performing guidance-free conditioning by feeding observations into rectified-flow dynamics while embedding hard physics via a constraint-preserving projection (e.g., incompressibility or conservation). Theoretically, we establish invariance guarantees to ensure that trajectories remain on the physics-consistent manifold throughout sampling. Experiments on various PDE systems demonstrate competitive reconstruction accuracy with sound physics consistency, while enabling efficient conditional sampling (e.g., 50 steps) and up to 320x faster inference than 2000-step guided diffusion baselines.

Hao Zhou, Rui Zhang, Han Wan +1

Gaoling School of Artificial Intelligence, Renmin University of China

Physics-Informed Generative ModelingUncertainty QuantificationRectified FlowGenerative ModelingPhysics-Informed MLFlow-Based Generative Modeling+1
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May 4, 2026

Generative Modeling with Orbit-Space Particle Flow Matching

We present Orbit-Space Geometric Probability Paths (OGPP), a particle-native flow-matching framework for generative modeling of particle systems. OGPP is motivated by two insights: (i) particles are defined up to permutation symmetries, so anonymous indexing inflates per-index target variance and yields curved, hard-to-learn flows; and (ii) particles live in physical space, so the flow terminal velocity has physical meaning and can encode geometric attributes, e.g., surface normals. OGPP instantiates three key components: (1) orbit-space canonicalization of the probability-path terminal endpoint, (2) particle index embeddings for role specialization, and (3) geometric probability paths with arc-length-aware terminal velocities that generate normals as a byproduct of the flow. We evaluate OGPP on minimal-surface benchmarks, where it reduces metric error by up to two orders of magnitude in a single inference step; on ShapeNet, where it matches the state of the art with 5x fewer steps and reaches airplane EMD comparable to DiT-3D with 26x fewer parameters and 5x fewer steps; and on single-shape encoding, where it produces normals and reconstructions competitive with 6D generators while operating entirely in 3D.

Sinan Wang, Jinjin He, Shenyifan Lu +3

Georgia Institute of Technology, USA

Flow MatchingOne-Step Generative ModelingGenerative Modeling3D Shape GenerationPermutation-Invariant LearningFlow-Based Generative Modeling
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May 3, 2026

Stable GFlowNets with Probabilistic Guarantees

Generative Flow Networks (GFlowNets) learn to sample states proportional to an unnormalized reward. Despite their theoretical promise, practical training is often unstable, exhibiting severe loss spikes and mode collapse. To tackle this, we first assess the sensitivity of GFlowNet objectives, demonstrating that a small Total Variation (TV) distance between the learned and target distributions does not preclude unbounded training loss. Motivated by this mismatch, we establish converse guarantees by deriving loss-to-TV bounds that certify global fidelity from bounded trajectory balance losses. Lastly, we propose Stable GFlowNets, an algorithm that leverages our theoretical results to stabilize training, and empirically demonstrate improved training behavior and superior distributional fidelity.

Zengxiang Lei, Ananth Shreekumar, Jonathan Rosenthal +6

Lyles School of Civil and Construction Engineering, Purdue University, IN, USA · Department of Computer Science, Purdue University, IN, USA · Computer Science and Engineering, University of California, Santa Cruz, USA

Generative Flow NetworksFlow-Based Generative Modeling
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May 2, 2026

CoFlow: Coordinated Few-Step Flow for Offline Multi-Agent Decision Making

Generative models have emerged as a promising paradigm for offline multi-agent reinforcement learning (MARL), but existing approaches require many iterative sampling steps. Recent few-step acceleration methods either distill a joint teacher into independent students or apply averaged velocity fields independently to each agent. Unfortunately, these few-step approaches hurt inter-agent coordination. We show that the efficiency-coordination trade-off is not inherent: single-pass multi-agent generation can preserve coordination when the velocity field is natively joint-coupled. We propose Coordinated few-step Flow (CoFlow), an architecture that combines Coordinated Velocity Attention (CVA) with Adaptive Coordination Gating. A finite-difference consistency surrogate further replaces memory-prohibitive Jacobian-vector product backpropagation through the averaged velocity field with two stop-gradient forward passes. Across 60 configurations spanning MPE, MA-MuJoCo, and SMAC, CoFlow matches or surpasses Gaussian policies, value-based methods, transformer policies, diffusion models, and prior flow baselines on episodic return. Three independent coordination probes confirm that CoFlow's improvements arise from inter-agent coordination rather than per-agent capacity. A denoising-step sweep shows that single-pass inference suffices on every configuration. CoFlow reaches state-of-the-art coordination quality in 1-3 denoising steps under both centralized and decentralized execution. Project Page: https://guowei-zou.github.io/coflow/

Guowei Zou, Haitao Wang, Beiwen Zhang +2

Sun Yat-sen University

Flow MatchingMulti-Agent CoordinationGenerative Flow NetworksFlow-Based Generative ModelingMulti-Agent Reinforcement LearningOffline MARL
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May 1, 2026

Posterior Augmented Flow Matching

Flow matching (FM) trains a time-dependent vector field that transports samples from a simple prior to a complex data distribution. However, for high-dimensional images, each training sample supervises only a single trajectory and intermediate point, yielding an extremely sparse and high-variance training signal. This under-constrained supervision can cause flow collapse, where the learned dynamics memorize specific source-target pairings, mapping diverse inputs to overly similar outputs, failing to generalize. We introduce Posterior-Augmented Flow Matching (PAFM), a theoretically grounded generalization of FM that replaces single-target supervision with an expectation over an approximate posterior of valid target completions for a given intermediate state and condition. PAFM factorizes this intractable posterior into (i) the likelihood of the intermediate under a hypothesized endpoint and (ii) the prior probability of that endpoint under the condition, and uses an importance sampling scheme to construct a mixture over multiple candidate targets. We prove that PAFM yields an unbiased estimator of the original FM objective while substantially reducing gradient variance during training by aggregating information from many plausible continuation trajectories per intermediate. Finally, we show that PAFM improves over FM by up to 3.4 FID50K across different model scales (SiT-B/2 and SiT-XL/2), different architectures (SiT and MMDiT), and in both class and text conditioned benchmarks (ImageNet and CC12M), with a negligible increase in the compute overhead. Code: https://github.com/gstoica27/PAFM.git.

George Stoica, Sayak Paul, Matthew Wallingford +6

Georgia Tech · University of Washington · Hugging Face +2

Flow MatchingConditional Image GenerationFlow-Based Generative ModelingVariance Reduction
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