Flow Matching

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Overview

Flow matching learns a velocity field that moves samples from a simple source, often Gaussian noise, toward a target data distribution. Conditional flow matching trains the field on paths between source and data samples, while optimal-transport couplings and rectified paths aim to make those trajectories easier to learn and integrate with an ordinary differential equation solver. Extensions handle discrete data, curved manifolds and symmetries, physical constraints, and applications from image and speech generation to robotics, scientific forecasting, and inverse problems. A central challenge is getting high fidelity and diverse, reliable samples with few solver steps; theory on convergence and stability is advancing, but the best paths, source distributions, and schedules depend on the task.

Latest papers

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663 papers123 papers in the last four weeks, up 68% on the four weeks before. 0.7% of all new papers.

Apr 16, 2026

One-Shot Generative Flows: Existence and Obstructions

We study dynamic measure transport for generative modeling, focusing on transport maps that connect a source measure P0P_0 to a target measure P1P_1 by integrating a velocity field of the form vt(x)=E[X˙t∣Xt=x]v_t(x) = \mathbb{E}[\dot X_t \mid X_t = x], where X∙=(Xt)tX_\bullet = (X_t)_t is a stochastic process satisfying (X0,X1)∼P0⊗P1(X_0,X_1)\sim{P_0}\otimes{P_1} and X˙t\dot X_t is its time derivative. We investigate when X∙X_\bullet induces a straight-line flow: a flow whose pointwise acceleration vanishes and is therefore exactly integrable by any first-order method. First, we develop multiple characterizations of straight-line flows in terms of PDEs involving the conditional statistics of the process. Then, we prove that straight-line flows under endpoint independence exhibit a sharp dichotomy. On the one hand, we construct explicit, computable straight-line processes for arbitrary Gaussian endpoints. On the other hand, we show that straight-line processes do not exist for targets with sufficiently well-separated modes. We demonstrate this obstruction through a sequence of increasingly general impossibility theorems that uncover a fundamental relationship between the sample-path behavior of a process with independent endpoints and the space-time geometry of this process' flow map. Taken together, these results provide a structural theory of when straight-line generative flows can, and cannot, exist.

Panos Tsimpos, Daniel Sharp, Youssef Marzouk

Operations Research Center · Massachusetts Institute of Technology · Cambridge, MA 02139 +2

Flow MatchingOne-Step Generative ModelingFlow-Based Generative ModelingOptimal Transport
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Apr 16, 2026

LeapAlign: Post-Training Flow Matching Models at Any Generation Step by Building Two-Step Trajectories

This paper focuses on the alignment of flow matching models with human preferences. A promising way is fine-tuning by directly backpropagating reward gradients through the differentiable generation process of flow matching. However, backpropagating through long trajectories results in prohibitive memory costs and gradient explosion. Therefore, direct-gradient methods struggle to update early generation steps, which are crucial for determining the global structure of the final image. To address this issue, we introduce LeapAlign, a fine-tuning method that reduces computational cost and enables direct gradient propagation from reward to early generation steps. Specifically, we shorten the long trajectory into only two steps by designing two consecutive leaps, each skipping multiple ODE sampling steps and predicting future latents in a single step. By randomizing the start and end timesteps of the leaps, LeapAlign leads to efficient and stable model updates at any generation step. To better use such shortened trajectories, we assign higher training weights to those that are more consistent with the long generation path. To further enhance gradient stability, we reduce the weights of gradient terms with large magnitude, instead of completely removing them as done in previous works. When fine-tuning the Flux model, LeapAlign consistently outperforms state-of-the-art GRPO-based and direct-gradient methods across various metrics, achieving superior image quality and image-text alignment.

Zhanhao Liang, Tao Yang, Jie Wu +2

1The Australian National University · 2ByteDance Seed

Diffusion Model AlignmentFlow MatchingImage GenerationDiffusion Model Fine-TuningFlow-Based Generative Modeling
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Apr 16, 2026

Towards Faster Language Model Inference Using Mixture-of-Experts Flow Matching

Flow matching retains the generation quality of diffusion models while enabling substantially faster inference, making it a compelling paradigm for generative modeling. However, when applied to language modeling, it exhibits fundamental limitations in representing complex latent distributions with irregular geometries, such as anisotropy and multimodality. To address these challenges, we propose a mixture-of-experts flow matching (MoE-FM) framework, which captures complex global transport geometries in latent space by decomposing them into locally specialized vector fields. Building on MoE-FM, we develop a non-autoregressive (NAR) language modeling approach, named YAN, instantiated with both Transformer and Mamba architectures. Across multiple downstream tasks, YAN achieves generation quality on par with both autoregressive (AR) and diffusion-based NAR language models, while requiring as few as three sampling steps. This yields a 40×40\times speedup over AR baselines and up to a 103×10^3\times speedup over diffusion language models, demonstrating substantial efficiency advantages for language modeling.

Aihua Li

Duke University

Non-Autoregressive Text GenerationFlow MatchingDiffusion Language Model InferenceFlow-Based Generative ModelingEfficient Language Model Inference
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Apr 15, 2026

PRiMeFlow: Capturing Complex Expression Heterogeneity in Perturbation Response Modelling

Predicting the effects of perturbations in-silico on cell state can identify drivers of cell behavior at scale and accelerate drug discovery. However, modeling challenges remain due to the inherent heterogeneity of single cell gene expression and the complex, latent gene dependencies. Here, we present PRiMeFlow, an end-to-end flow matching based approach to directly model the effects of genetic and small molecule perturbations in the gene expression space. The distribution-fitting approach taken by PRiMeFlow enables it to accurately approximate the empirical distribution of single-cell gene expression, which we demonstrate through extensive benchmarking inside PerturBench. Through ablation studies, we also validate important model design choices such as operating in gene expression space and parameterizing the velocity field with a U-Net architecture. Finally, by scaling PRiMeFlow to a broad perturbation data atlas spanning multiple datasets and employing a carefully designed pretraining-finetuning strategy, we demonstrate its outstanding performance on the H1 human embryonic stem cells from the ARC Virtual Cell Challenge benchmark.

Zichao Yan, Yan Wu, Mica Xu Ji +10

Altos Labs · Current affiliation: Verge Genomics · Current affiliation: Loka +1

Single-Cell Perturbation ModelingSingle-Cell TranscriptomicsFlow-Based Generative ModelingFlow Matching
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Apr 15, 2026

EMGFlow: Robust and Efficient Surface Electromyography Synthesis via Flow Matching

Deep learning-based surface electromyography (sEMG) gesture recognition is frequently bottlenecked by data scarcity and limited subject diversity. While synthetic data generation via Generative Adversarial Networks (GANs) and diffusion models has emerged as a promising augmentation strategy, these approaches often face challenges regarding training stability or inference efficiency. To bridge this gap, we propose EMGFlow, a conditional sEMG generation framework. To the best of our knowledge, this is the first study to investigate the application of Flow Matching (FM) and continuous-time generative modeling in the sEMG domain. To validate EMGFlow across three benchmark sEMG datasets, we employ a unified evaluation protocol integrating feature-based fidelity, distributional geometry, and downstream utility. Extensive evaluations show that EMGFlow outperforms conventional augmentation and GAN baselines, and provides stronger standalone utility than the diffusion baselines considered here under the train-on-synthetic test-on-real (TSTR) protocol. Furthermore, by optimizing generation dynamics through advanced numerical solvers and targeted time sampling, EMGFlow achieves improved quality-efficiency trade-offs. Taken together, these results suggest that Flow Matching is a promising and efficient paradigm for addressing data bottlenecks in myoelectric control systems. Our code is available at: https://github.com/Open-EXG/EMGFlow.

Boxuan Jiang, Chenyun Dai, Can Han

School of Biomedical Engineering, Shanghai Jiao Tong University, Shanghai, 200240, China

Synthetic Data AugmentationConditional Generative ModelingFlow-Based Generative ModelingFlow Matching
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Apr 14, 2026

Self-Adversarial One Step Generation via Condition Shifting

The push for efficient text to image synthesis has moved the field toward one step sampling, yet existing methods still face a three way tradeoff among fidelity, inference speed, and training efficiency. Approaches that rely on external discriminators can sharpen one step performance, but they often introduce training instability, high GPU memory overhead, and slow convergence, which complicates scaling and parameter efficient tuning. In contrast, regression based distillation and consistency objectives are easier to optimize, but they typically lose fine details when constrained to a single step. We present APEX, built on a key theoretical insight: adversarial correction signals can be extracted endogenously from a flow model through condition shifting. Using a transformation creates a shifted condition branch whose velocity field serves as an independent estimator of the model's current generation distribution, yielding a gradient that is provably GAN aligned, replacing the sample dependent discriminator terms that cause gradient vanishing. This discriminator free design is architecture preserving, making APEX a plug and play framework compatible with both full parameter and LoRA based tuning. Empirically, our 0.6B model surpasses FLUX-Schnell 12B (20×\times more parameters) in one step quality. With LoRA tuning on Qwen-Image 20B, APEX reaches a GenEval score of 0.89 at NFE=1 in 6 hours, surpassing the original 50-step teacher (0.87) and providing a 15.33×\times inference speedup. Code is available https://github.com/LINs-lab/APEX.

Deyuan Liu, Peng Sun, Yansen Han +3

Westlake University · Zhejiang University · Shanghai Innovation Institute +1

One-Step Generative ModelingT2I Diffusion ModelsAdversarial TrainingFlow Matching
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Apr 14, 2026

SubFlow: Sub-mode Conditioned Flow Matching for Diverse One-Step Generation

Flow matching has emerged as a powerful generative framework, with recent few-step methods achieving remarkable inference acceleration. However, we identify a critical yet overlooked limitation: these models suffer from severe diversity degradation, concentrating samples on dominant modes while neglecting rare but valid variations of the target distribution. We trace this degradation to averaging distortion: when trained with MSE objectives, class-conditional flows learn a frequency-weighted mean over intra-class sub-modes, causing the model to over-represent high-density modes while systematically neglecting low-density ones. To address this, we propose SubFlow, Sub-mode Conditioned Flow Matching, which eliminates averaging distortion by decomposing each class into fine-grained sub-modes via semantic clustering and conditioning the flow on sub-mode indices. Each conditioned sub-distribution is approximately unimodal, so the learned flow accurately targets individual modes with no averaging distortion, restoring full mode coverage in a single inference step. Crucially, SubFlow is entirely plug-and-play: it integrates seamlessly into existing one-step models such as MeanFlow and Shortcut Models without any architectural modifications. Extensive experiments on ImageNet-256 demonstrate that SubFlow yields substantial gains in generation diversity (Recall) while maintaining competitive image quality (FID), confirming its broad applicability across different one-step generation frameworks. Project page: https://yexionglin.github.io/subflow.

Yexiong Lin, Jia Shi, Shanshan Ye +3

Sydney AI Centre, The University of Sydney, Sydney, NSW, Australia · School of Artificial Intelligence, Xidian University, Xi’an, Shaanxi, China · Australian Artificial Intelligence Institute, University of Technology Sydney, Sydney, NSW, Australia +1

Flow MatchingFlow-Based Generative ModelingOne-Step Generative ModelingConditional Generative ModelingMode Collapse
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Apr 13, 2026

LangFlow: Continuous Diffusion Rivals Discrete in Language Modeling

Continuous diffusion has been the foundation of high-fidelity, controllable, and few-step generation of many data modalities such as images. However, in language modeling, prior continuous diffusion language models (DLMs) lag behind discrete counterparts due to the sparse data space and the underexplored design space. In this work, we close this gap with LangFlow, the first continuous DLM to rival discrete diffusion, by connecting embedding-space DLMs to Flow Matching via Bregman divergence, alongside three key innovations: (1) we derive a novel ODE-based NLL bound for principled evaluation of continuous flow-based language models; (2) we propose an information-uniform principle for setting the noise schedule, which motivates a learnable noise scheduler based on a Gumbel distribution; and (3) we revise prior training protocols by incorporating self-conditioning, as we find it improves both likelihood and sample quality of embedding-space DLMs with effects substantially different from discrete diffusion. Putting everything together, LangFlow rivals top discrete DLMs on both the perplexity (PPL) and the generative perplexity (Gen. PPL), reaching a PPL of 30.0 on LM1B and 24.6 on OpenWebText. It even exceeds autoregressive baselines in zero-shot transfer on 4 out of 7 benchmarks. LangFlow provides the first clear evidence that continuous diffusion is a promising paradigm for language modeling. Homepage: https://github.com/nealchen2003/LangFlow

Yuxin Chen, Chumeng Liang, Hangke Sui +4

UIUC

Self-ConditioningDiffusion Language ModelsContinuous Diffusion Language ModelsDiffusion Language Model TrainingFlow Matching
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Apr 13, 2026

LayerCache: Exploiting Layer-wise Velocity Heterogeneity for Efficient Flow Matching Inference

Flow Matching models achieve state-of-the-art image generation quality but incur substantial inference cost due to iterative denoising through large Transformer networks. We observe that different layer groups within a Transformer exhibit markedly heterogeneous velocity dynamics: shallow layers are highly stable and amenable to aggressive caching, while deep layers undergo large velocity changes that demand full computation. Existing caching methods, however, treat the entire Transformer as a monolithic unit, applying a single caching decision per timestep and thus failing to exploit this heterogeneity. Based on this finding, we propose LayerCache, a layer-aware caching framework that partitions the Transformer into layer groups and makes independent, per-group caching decisions at each denoising step. LayerCache introduces an adaptive JVP span K selection mechanism that leverages per-group stability measurements to balance estimation accuracy and computational savings. We formulate a three-dimensional scheduling problem over timesteps, layer groups, and JVP span, and solve it with a greedy budget allocation algorithm. On Qwen-Image (1024x1024, 50 steps), LayerCache achieves PSNR 37.46 dB (+5.38 dB over MeanCache), SSIM 0.9834, and LPIPS 0.0178 (a 70% reduction over MeanCache) at 1.37x speedup, dominating all prior caching methods on the quality-speed Pareto frontier.

Guandong Li

iFLYTEK

Flow MatchingFlow-Based Generative ModelingDiffusion Transformer
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Apr 10, 2026

NeuroFlow: Toward Unified Visual Encoding and Decoding from Neural Activity

Visual encoding and decoding models act as gateways to understanding the neural mechanisms underlying human visual perception. Typically, visual encoding models that predict brain activity from stimuli and decoding models that reproduce stimuli from brain activity are treated as distinct tasks, requiring separate models and training procedures. This separation is inefficient and fails to model the consistency between encoding and decoding processes. To address this limitation, we propose NeuroFlow, the first unified framework that jointly models visual encoding and decoding from neural activity within a single flow model. NeuroFlow introduces two key components: (1) NeuroVAE is designed as a variational backbone to model neural variability and establish a compact, semantically structured latent space for bidirectional modeling across visual and neural modalities. (2) Cross-modal Flow Matching (XFM) bypasses the typical paradigm of noise-to-data diffusion guided by a specific modality condition, instead learning a reversibly consistent flow model between visual and neural latent distributions. For the first time, visual encoding and decoding are reformulated as a time-dependent, reversible process within a shared latent space for unified modeling. Empirical results demonstrate that NeuroFlow achieves superior overall performance in visual encoding and decoding tasks with higher computational efficiency compared to any isolated methods. We further analyze principal factors that steer the model toward encoding-decoding consistency and, through brain functional analyses, demonstrate that NeuroFlow captures consistent activation patterns underlying neural variability. NeuroFlow marks a major step toward unified visual encoding and decoding from neural activity, providing mechanistic insights that inform future bidirectional visual brain-computer interfaces.

Weijian Mai, Mu Nan, Yu Zhu +6

Shanghai Artificial Intelligence Laboratory · University of Hong Kong · Shenzhen Loop Area Institute +2

Neural DecodingNeural EncodingVisual NeuroscienceFlow Matching
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Apr 10, 2026

Efficient Hierarchical Implicit Flow Q-learning for Offline Goal-conditioned Reinforcement Learning

Offline goal-conditioned reinforcement learning (GCRL) is a practical reinforcement learning paradigm that aims to learn goal-conditioned policies from reward-free offline data. Despite recent advances in hierarchical architectures such as HIQL, long-horizon control in offline GCRL remains challenging due to the limited expressiveness of Gaussian policies and the inability of high-level policies to generate effective subgoals. To address these limitations, we propose the goal-conditioned mean flow policy, which introduces an average velocity field into hierarchical policy modeling for offline GCRL. Specifically, the mean flow policy captures complex target distributions for both high-level and low-level policies through a learned average velocity field, enabling efficient action generation via one-step sampling. Furthermore, considering the insufficiency of goal representation, we introduce a LeJEPA loss that repels goal representation embeddings during training, thereby encouraging more discriminative representations and improving generalization. Experimental results show that our method achieves strong performance across both state-based and pixel-based tasks in the OGBench benchmark.

Zhiqiang Dong, Teng Pang, Rongjian Xu +1

School of Software, Shandong University, Jinan, Shandong, China · Luke EI, Jinan, China

Offline RLHierarchical RLRepresentation Learning for RLFlow Matching
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Apr 9, 2026

Equivariant Efficient Joint Discrete and Continuous MeanFlow for Molecular Graph Generation

Graph-structured data jointly contain discrete topology and continuous geometry, which poses fundamental challenges for generative modeling due to heterogeneous distributions, incompatible noise dynamics, and the need for equivariant inductive biases. Existing flow-matching approaches for graph generation typically decouple structure from geometry, lack synchronized cross-domain dynamics, and rely on iterative sampling, often resulting in physically inconsistent molecular conformations and slow sampling. To address these limitations, we propose Equivariant MeanFlow (EQUIMF), a unified SE(3)-equivariant generative framework that jointly models discrete and continuous components through synchronized MeanFlow dynamics. EQUIMF introduces a unified time bridge and average-velocity updates with mutual conditioning between structure and geometry, enabling efficient few-step generation while preserving physical consistency. Moreover, we develop a novel discrete MeanFlow formulation with a simple yet effective parameterization to support efficient generation over discrete graph structures. Extensive experiments demonstrate that EQUIMF consistently outperforms prior diffusion and flow-matching methods in generation quality, physical validity, and sampling efficiency.

Rongjian Xu, Teng Pang, Zhiqiang Dong +1

School of Software, Shandong University, Jinan, Shandong, China · School of Software, Shandong University, Jinan, Shandong, China; Luke EI, Jinan, China

Molecular Graph GenerationGenerative Modeling3D Molecular GenerationFlow-Based Generative ModelingFlow MatchingDiscrete Flow Matching
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Apr 9, 2026

Value-Guidance MeanFlow for Offline Multi-Agent Reinforcement Learning

Offline multi-agent reinforcement learning (MARL) aims to learn the optimal joint policy from pre-collected datasets, requiring a trade-off between maximizing global returns and mitigating distribution shift from offline data. Recent studies use diffusion or flow generative models to capture complex joint policy behaviors among agents; however, they typically rely on multi-step iterative sampling, thereby reducing training and inference efficiency. Although further research improves sampling efficiency through methods like distillation, it remains sensitive to the behavior regularization coefficient. To address the above-mentioned issues, we propose Value Guidance Multi-agent MeanFlow Policy (VGM2^2P), a simple yet effective flow-based policy learning framework that enables efficient action generation with coefficient-insensitive conditional behavior cloning. Specifically, VGM2^2P uses global advantage values to guide agent collaboration, treating optimal policy learning as conditional behavior cloning. Additionally, to improve policy expressiveness and inference efficiency in multi-agent scenarios, it leverages classifier-free guidance MeanFlow for both policy training and execution. Experiments on tasks with both discrete and continuous action spaces demonstrate that, even when trained solely via conditional behavior cloning, VGM2^2P efficiently achieves performance comparable to state-of-the-art methods.

Teng Pang, Zhiqiang Dong, Yan Zhang +3

School of Software, Shandong University, Jinan, China · Luke EI, Jinan, China

Offline MARLCooperative MARLFlow Matching
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Apr 7, 2026

ODE-free Neural Flow Matching for One-Step Generative Modeling

Diffusion and flow matching models generate samples by learning time-dependent vector fields whose integration transports noise to data, requiring tens to hundreds of network evaluations at inference. We instead learn the transport map directly. We propose Optimal Transport Neural Flow Matching (OT-NFM), an ODE-free generative framework that parameterizes the flow map with neural flows, enabling true one-step generation with a single forward pass. We show that naive flow-map training suffers from mean collapse, where inconsistent noise-data pairings drive all outputs toward the data mean. We prove that consistent coupling is necessary for non-degenerate learning and address this using optimal transport pairings with scalable minibatch and online coupling strategies. Experiments on synthetic benchmarks and image generation tasks (MNIST and CIFAR-10) demonstrate competitive sample quality while reducing inference to a single network evaluation.

Xiao Shou

Baylor University

One-Step Generative ModelingFlow MatchingFlow Map LearningImage Generation
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Apr 7, 2026

Optimal-Transport-Guided Functional Flow Matching for Turbulent Field Generation in Hilbert Space

High-fidelity modeling of turbulent flows requires capturing complex spatiotemporal dynamics and multi-scale intermittency, posing a fundamental challenge for traditional knowledge-based systems. While deep generative models, such as diffusion models and Flow Matching, have shown promising performance, they are fundamentally constrained by their discrete, pixel-based nature. This limitation restricts their applicability in turbulence computing, where data inherently exists in a functional form. To address this gap, we propose Functional Optimal Transport Conditional Flow Matching (FOT-CFM), a generative framework defined directly in infinite-dimensional function space. Unlike conventional approaches defined on fixed grids, FOT-CFM treats physical fields as elements of an infinite-dimensional Hilbert space, and learns resolution-invariant generative dynamics directly at the level of probability measures. By integrating Optimal Transport (OT) theory, we construct deterministic, straight-line probability paths between noise and data measures in Hilbert space. This formulation enables simulation-free training and significantly accelerates the sampling process. We rigorously evaluate the proposed system on a diverse suite of chaotic dynamical systems, including the Navier-Stokes equations, Kolmogorov Flow, and Hasegawa-Wakatani equations, all of which exhibit rich multi-scale turbulent structures. Experimental results demonstrate that FOT-CFM achieves superior fidelity in reproducing high-order turbulent statistics and energy spectra compared to state-of-the-art baselines.

Li Kunpeng, Wan Chenguang, Qu Zhisong +5

Flow-Based Generative ModelingFlow MatchingComputational Fluid DynamicsOptimal Transport
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Apr 7, 2026

SnapFlow: One-Step Action Generation for Flow-Matching VLAs via Progressive Self-Distillation

Vision-language-action (VLA) models with flow-matching action heads, such as π0 and π0.5, have become a standard recipe for generalist robot manipulation, but every action chunk is produced by integrating a learned velocity field over about ten sequential action-expert evaluations, so action generation dominates model latency and caps the achievable control frequency. Existing remedies either shrink each evaluation through layer distillation or pruning while leaving the sequential solver in place, or import one-step objectives from image generation that were designed for training from scratch, rely on an exponential-moving-average teacher or an external critic, and regress finite-span targets onto the conditional velocity, whose variance around the marginal field biases the learned map. We introduce SnapFlow, a self-distillation method that adapts a pretrained flow-matching VLA to one-step action generation. SnapFlow mixes local flow-matching supervision with shortcut samples whose target is the model's own two-step Euler endpoint under stop-gradient, and adds a single zero-initialized destination-time projection so that one action expert serves both objectives while the pretrained predictor is preserved at initialization; no separate teacher network or critic is required. On π0.5, whose 10-NFE teacher operates near ceiling, the archived one-NFE SnapFlow checkpoint reaches 98.75% LIBERO success versus 97.75% for the teacher; under a separate paired evaluation of π0, the one-NFE model surpasses its teacher by 10.5 percentage points. SnapFlow also lowers mean and tail action error on held-out data, cuts batch-one model latency by about 3.3x, and transfers to shelf retrieval on a dual-arm humanoid.

Wuyang Luan, Junhui Li, Weiguang Zhao +2

Jilin University · Chongqing University · University of Liverpool +1

Efficient VLA Model InferenceVision-Language-Action ModelsFlow MatchingSelf-DistillationRobotic Manipulation
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Apr 7, 2026

Attention-Guided Flow-Matching for Sparse 3D Geological Generation

Constructing high-resolution 3D geological models from sparse 1D borehole and 2D surface data is a highly ill-posed inverse problem. Traditional heuristic and implicit modeling methods fundamentally fail to capture non-linear topological discontinuities under extreme sparsity, often yielding unrealistic artifacts. Furthermore, while deep generative architectures like Diffusion Models have revolutionized continuous domains, they suffer from severe representation collapse when conditioned on sparse categorical grids. To bridge this gap, we propose 3D-GeoFlow, the first Attention-Guided Continuous Flow Matching framework tailored for sparse multimodal geological modeling. By reformulating discrete categorical generation as a simulation-free, continuous vector field regression optimized via Mean Squared Error, our model establishes stable, deterministic optimal transport paths. Crucially, we integrate 3D Attention Gates to dynamically propagate localized borehole features across the volumetric latent space, ensuring macroscopic structural coherence. To validate our framework, we curated a large-scale multimodal dataset comprising 2,200 procedurally generated 3D geological cases. Extensive out-of-distribution (OOD) evaluations demonstrate that 3D-GeoFlow achieves a paradigm shift, significantly outperforming heuristic interpolations and standard diffusion baselines.

Zhixiang Lu, Mengqi Han, Peixin Guo +4

Xi’an Jiaotong-Liverpool University, Jiangsu 215123, China · Deep Optica, Shanghai 200030, China · University of Chinese Academy of Sciences, Beijing 100049, China

Flow-Based Generative ModelingFlow Matching
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Apr 6, 2026

Training-Free Refinement of Flow Matching with Divergence-based Sampling

Flow-based models learn a target distribution by modeling a marginal velocity field, defined as the average of sample-wise velocities connecting each sample from a simple prior to the target data. When sample-wise velocities conflict at the same intermediate state, however, this averaged velocity can misguide samples toward low-density regions, degrading generation quality. To address this issue, we propose the Flow Divergence Sampler (FDS), a training-free framework that refines intermediate states before each solver step. Our key finding reveals that the severity of this misguidance is quantified by the divergence of the marginal velocity field that is readily computable during inference with a well-optimized model. FDS exploits this signal to steer states toward less ambiguous regions. As a plug-and-play framework compatible with standard solvers and off-the-shelf flow backbones, FDS consistently improves fidelity across various generation tasks including text-to-image synthesis, and inverse problems.

Yeonwoo Cha, Jaehoon Yoo, Semin Kim +3

KAIST

Flow MatchingT2I GenerationFlow-Based Generative Modeling
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Apr 6, 2026

Isokinetic Flow Matching for Pathwise Straightening of Generative Flows

Flow Matching (FM) constructs linear conditional probability paths, but the learned marginal velocity field inevitably exhibits strong curvature due to trajectory superposition. This curvature severely inflates numerical truncation errors, bottlenecking few-step sampling. To overcome this, we introduce Isokinetic Flow Matching (Iso-FM), a lightweight, Jacobian-free dynamical regularizer that directly penalizes pathwise acceleration. By using a self-guided finite-difference approximation of the material derivative Dv/Dt, Iso-FM enforces local velocity consistency without requiring auxiliary encoders or expensive second-order autodifferentiation. Operating as a pure plug-and-play addition to single-stage FM training, Iso-FM dramatically improves few-step generation. On CIFAR-10 (DiT-S/2), Iso-FM slashes conditional non-OT FID at 2 steps from 78.82 to 27.13 - a 2.9x relative efficiency gain - and reaches a best-observed FID at 4 steps of 10.23. These results firmly establish acceleration regularization as a principled, compute-efficient mechanism for fast generative sampling.

Tauhid Khan

Flow MatchingFlow-Based Generative ModelingFew-Step Diffusion Sampling
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Apr 5, 2026

Correcting Source Mismatch in Flow Matching with Radial-Angular Transport

Flow Matching is typically built from Gaussian sources and Euclidean probability paths. For heavy-tailed or anisotropic data, however, a Gaussian source induces a structural mismatch already at the level of the radial distribution. We introduce Radial--Angular Flow Matching (RAFM), a framework that explicitly corrects this source mismatch within the standard simulation-free Flow Matching template. RAFM uses a source whose radial law matches that of the data and whose conditional angular distribution is uniform on the sphere, thereby removing the Gaussian radial mismatch by construction. This reduces the remaining transport problem to angular alignment, which leads naturally to conditional paths on scaled spheres defined by spherical geodesic interpolation. The resulting framework yields explicit Flow Matching targets tailored to radial--angular transport without modifying the underlying deterministic training pipeline. We establish the exact density of the matched-radial source, prove a radial--angular KL decomposition that isolates the Gaussian radial penalty, characterize the induced target vector field, and derive a stability result linking Flow Matching error to generation error. We further analyze empirical estimation of the radial law, for which Wasserstein and CDF metrics provide natural guarantees. Empirically, RAFM substantially improves over standard Gaussian Flow Matching and remains competitive with recent non-Gaussian alternatives while preserving a lightweight deterministic training procedure. Overall, RAFM provides a principled source-and-path design for Flow Matching on heavy-tailed and extreme-event data.

Fouad Oubari, Mathilde Mougeot

Université Paris-Saclay, CNRS, ENS Paris-Saclay, Centre Borelli · ENSIIE

Flow MatchingGenerative Modeling
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Apr 5, 2026

OP-GRPO: Efficient Off-Policy GRPO for Flow-Matching Models

Post training via GRPO has demonstrated remarkable effectiveness in improving the generation quality of flow-matching models. However, GRPO suffers from inherently low sample efficiency due to its on-policy training paradigm. To address this limitation, we present OP-GRPO, the first Off-Policy GRPO framework tailored for flow-matching models. First, we actively select high-quality trajectories and adaptively incorporate them into a replay buffer for reuse in subsequent training iterations. Second, to mitigate the distribution shift introduced by off-policy samples, we propose a sequence-level importance sampling correction that preserves the integrity of GRPO's clipping mechanism while ensuring stable policy updates. Third, we theoretically and empirically show that late denoising steps yield ill-conditioned off-policy ratios, and mitigate this by truncating trajectories at late steps. Across image and video generation benchmarks, OP-GRPO achieves comparable or superior performance to Flow-GRPO with only 34.2% of the training steps on average, yielding substantial gains in training efficiency while maintaining generation quality.

Liyu Zhang, Kehan Li, Tingrui Han +4

College of Control Science and Engineering, Zhejiang University · Central Research Institute, Huawei

RL for Generative ModelsOff-Policy RLFlow MatchingImage GenerationVideo GenerationExperience Replay
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Apr 5, 2026

TORA: Topological Representation Alignment for 3D Shape Assembly

Flow-matching methods for 3D shape assembly learn point-wise velocity fields that transport parts toward assembled configurations, yet they receive no explicit guidance about which cross-part interactions should drive the motion. We introduce TORA, a topology-first representation alignment framework that distills relational structure from a frozen pretrained 3D encoder into the flow-matching backbone during training. We first realize this via simple instantiation, token-wise cosine matching, which injects the learned geometric descriptors from the teacher representation. We then extend to employ a Centered Kernel Alignment (CKA) loss to match the similarity structure between student and teacher representations for enhanced topological alignment. Through systematic probing of diverse 3D encoders, we show that geometry- and contact-centric teacher properties, not semantic classification ability, govern alignment effectiveness, and that alignment is most beneficial at later transformer layers where spatial structure naturally emerges. TORA introduces zero inference overhead while yielding two consistent benefits: faster convergence (up to 6.9×\times) and improved accuracy in-distribution, along with greater robustness under domain shift. Experiments on five benchmarks spanning geometric, semantic, and inter-object assembly demonstrate state-of-the-art performance, with particularly pronounced gains in zero-shot transfer to unseen real-world and synthetic datasets. Project page: https://nahyuklee.github.io/tora.

Nahyuk Lee, Zhiang Chen, Marc Pollefeys +1

Independent Researcher · ETH Zurich · ETH AI Center

Flow Matching3D Shape Generation3D Representation LearningKnowledge Distillation
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Apr 4, 2026

From Euler to Dormand-Prince: ODE Solvers for Flow Matching Generative Models

Sampling from Flow Matching generative models requires solving an ordinary differential equation (ODE) whose computational cost is dominated by neural network forward passes. We derive four classical ODE solvers -- Euler, Explicit Midpoint, Classical Runge-Kutta (RK4), and Dormand-Prince 5(4) -- from first principles via Taylor expansion, implement them from scratch in PyTorch, and systematically benchmark their efficiency on Conditional Flow Matching tasks ranging from 2D toy distributions to MNIST digits. On the quantitative side, we use sliced Wasserstein distance to construct NFE-quality Pareto frontiers,finding that RK4 at 80 function evaluations achieves sample quality comparable to Euler at 200. Beyond reproducing known convergence rates, we report two empirical observations: (1) the Jacobian eigenvalue spectrum of the learned velocity field stiffens sharply near t=1, explaining why the adaptive Dormand-Prince solver automatically concentrates its step budget at the end of the trajectory; (2) the quality gap between low-order and high-order solvers widens for undertrained and smaller models, indicating that solver choice matters most when the model is imperfect. Code and all experiment scripts are publicly available.

Hao Xiao

ATLAS AI Lab

Numerical IntegrationFlow MatchingConditional Flow Matching
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Apr 3, 2026

Physics-Constrained Adaptive Flow Matching for Climate Downscaling

Regional climate information at kilometer scales is essential for assessing the impacts of climate change, but generating it with global climate models is too expensive due to their high computational costs. Machine learning models offer a fast alternative, yet they often violate basic physical laws and degrade when applied to climates outside of their training distribution. We present Physics-Constrained Adaptive Flow Matching (PC-AFM), a generative downscaling model that addresses both problems. Building on the Adaptive Flow Matching (AFM) model of Fotiadis et al. (2025) as our baseline, we add soft conservation constraints that keep the downscaled output consistent with the large-scale input for precipitation and humidity, and use gradient surgery via the ConFIG algorithm to prevent these constraints from interfering with the generative objective. We train the model on Central Europe climate data, evaluate it on a 10-time downscaling task (63km to 6.3km) over six variables (near-surface temperature, precipitation, specific humidity, surface pressure, and horizontal wind components) across a comprehensive set of metrics including bias, ensemble skill scores, power spectra, and conservation error, and test the generalization on two held-out climate regions. Within the training distribution, PC-AFM reduces conservation errors and improves ensemble calibration while matching the baseline on standard skill metrics. Outside the training distribution, where unconstrained models develop large systematic errors by extrapolating learned statistics, PC-AFM halves precipitation wet bias, reduces conservation error and improves extreme-quantile accuracy, all without any information about the target climate at inference time. These results indicate that physical consistency is a practical requirement for deploying generative downscaling models in real-world applications.

Kevin Debeire, Aytaç Paçal, Pierre Gentine +3

Deutsches Zentrum für Luft- und Raumfahrt (DLR), Institut für Physik der Atmosphäre, Oberpfaffenhofen, Germany · School of Computation, Information and Technology, Technical University of Munich, Germany · LEAP Center, Columbia University, New York, NY, 10027, USA +1

Climate DownscalingPhysics-Informed Generative ModelingConstrained Generative ModelingPrecipitation DownscalingFlow MatchingOOD Generalization
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Apr 3, 2026

A Flow Matching Framework for Soft-Robot Inverse Dynamics

Learning the inverse dynamics of soft continuum robots remains challenging due to high-dimensional nonlinearities and complex actuation coupling. Conventional feedback-based controllers often suffer from control chattering due to corrective oscillations, while deterministic regression-based learners struggle to capture the complex nonlinear mappings required for accurate dynamic tracking. Motivated by these limitations, we propose an inverse-dynamics framework for open-loop feedforward control that learns the system's differential dynamics as a generative transport map. Specifically, inverse dynamics is reformulated as a conditional flow-matching problem, and Rectified Flow (RF) is adopted as a lightweight instance to generate physically consistent control inputs rather than conditional averages. Two variants are introduced to further enhance physical consistency: RF-Physical, utilizing a physics-based prior for residual modeling; and RF-FWD, integrating a forward-dynamics consistency loss during flow matching. Extensive evaluations demonstrate that our framework reduces trajectory tracking RMSE by over 50% compared to standard regression baselines (MLP, LSTM, Transformer). The system sustains stable open-loop execution at a peak end-effector velocity of 1.14 m/s with sub-millisecond inference latency (0.995 ms). This work demonstrates flow matching as a robust, high-performance paradigm for learning differential inverse dynamics in soft robotic systems.

Hang Yang, Fangju Yang, Yangming Zhang +5

Robot Dynamics LearningInverse DynamicsSoft RoboticsFlow Matching
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Apr 3, 2026

Few-Shot Distribution-Aligned Flow Matching for Data Synthesis in Medical Image Segmentation

Data heterogeneity hinders clinical deployment of medical image analysis models, and generative data augmentation helps mitigate this issue. However, recent diffusion-based methods that synthesize image-mask pairs often ignore distribution shifts between generated and real images across scenarios, and such mismatches can markedly degrade downstream performance. To address this issue, we propose AlignFlow, a flow matching model that aligns with the target reference image distribution via differentiable reward fine-tuning, and remains effective even when only a small number of reference images are provided. Specifically, we divide the training of the flow matching model into two stages: in the first stage, the model fits the training data to generate plausible images; Then, we introduce a distribution alignment mechanism and employ differentiable reward to steer the generated images toward the distribution of the given samples from the target domain. In addition, to enhance the diversity of generated masks, we also design a flow matching based mask generation to complement the diversity in regions of interest. Extensive experiments demonstrate the effectiveness of our approach, i.e., performance improvement by 3.5-4.0% in mDice and 3.5-5.6% in mIoU across a variety of datasets and scenarios.

Jie Yang, Ziqi Ye, Aihua Ke +3

Key Laboratory of Aerospace Information Security and Trusted Computing, Ministry of Education, School of Cyber Science and Engineering, Wuhan University, Wuhan, China · Shanghai Innovation Institute, Shanghai, China · Shanghai AI Laboratory, Shanghai, China

Synthetic Data AugmentationMedical Image SynthesisDistribution MatchingFlow MatchingMedical Image Segmentation
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Apr 2, 2026

Beyond Fixed Inference: Quantitative Flow Matching for Adaptive Image Denoising

Diffusion and flow-based generative models have shown strong potential for image restoration. However, image denoising under unknown and varying noise conditions remains challenging, because the learned vector fields may become inconsistent across different noise levels, leading to degraded restoration quality under mismatch between training and inference. To address this issue, we propose a quantitative flow matching framework for adaptive image denoising. The method first estimates the input noise level from local pixel statistics, and then uses this quantitative estimate to adapt the inference trajectory, including the starting point, the number of integration steps, and the step-size schedule. In this way, the denoising process is better aligned with the actual corruption level of each input, reducing unnecessary computation for lightly corrupted images while providing sufficient refinement for heavily degraded ones. By coupling quantitative noise estimation with noise-adaptive flow inference, the proposed method improves both restoration accuracy and inference efficiency. Extensive experiments on natural, medical, and microscopy images demonstrate its robustness and strong generalization across diverse noise levels and imaging conditions.

Jigang Duan, Genwei Ma, Xu Jiang +3

School of Mathematical Sciences, Capital Normal University, Beijing 100048, China · National Center for Applied Mathematics Beijing, Capital Normal University and Academy for multidisciplinary studies, Capital Normal University, Beijing, 100048, China · Institute of Nuclear and New Energy Technology, Tsinghua University, Beijing 100084, China

Image DenoisingFlow MatchingFlow-Based Generative Modeling
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Mar 31, 2026

SANA I2I: A Text Free Flow Matching Framework for Paired Image to Image Translation with a Case Study in Fetal MRI Artifact Reduction

We propose SANA-I2I, a text-free high-resolution image-to-image generation framework that extends the SANA family by removing textual conditioning entirely. In contrast to SanaControlNet, which combines text and image-based control, SANA-I2I relies exclusively on paired source-target images to learn a conditional flow-matching model in latent space. The model learns a conditional velocity field that maps a target image distribution to another one, enabling supervised image translation without reliance on language prompts. We evaluate the proposed approach on the challenging task of fetal MRI motion artifact reduction. To enable paired training in this application, where real paired data are difficult to acquire, we adopt a synthetic data generation strategy based on the method proposed by Duffy et al., which simulates realistic motion artifacts in fetal magnetic resonance imaging (MRI). Experimental results demonstrate that SANA-I2I effectively suppresses motion artifacts while preserving anatomical structure, achieving competitive performance few inference steps. These results highlight the efficiency and suitability of our proposed flow-based, text-free generative models for supervised image-to-image tasks in medical imaging.

Italo Felix Santos, Gilson Antonio Giraldi, Heron Werner Junior

National Laboratory for Scientific Computing - LNCC · Biodesign Laboratory Dasa

Image-to-Image TranslationConditional Image GenerationFlow MatchingMagnetic Resonance Imaging
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Mar 31, 2026

WorldFlow3D: Flowing Through 3D Distributions for Unbounded World Generation

Unbounded 3D world generation is emerging as a foundational task for scene modeling in computer vision, graphics, and robotics. In this work, we present WorldFlow3D, a novel method capable of generating unbounded 3D worlds. Building upon a foundational property of flow matching - namely, defining a path of transport between two data distributions - we model 3D generation more generally as a problem of flowing through 3D data distributions, not limited to conditional denoising. We find that our latent-free flow approach generates causal and accurate 3D structure, and can use this as an intermediate distribution to guide the generation of more complex structure and high-quality texture - all while converging more rapidly than existing methods. We enable controllability over generated scenes with vectorized scene layout conditions for geometric structure control and visual texture control through scene attributes. We confirm the effectiveness of WorldFlow3D on both real outdoor driving scenes and synthetic indoor scenes, validating cross-domain generalizability and high-quality generation on real data distributions. We confirm favorable scene generation fidelity over approaches in all tested settings for unbounded scene generation. For more, see https://light.princeton.edu/worldflow3d.

Amogh Joshi, Julian Ost, Felix Heide

Princeton University · Torc Robotics

3D Scene Generation3D Scene Layout GenerationFlow Matching
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Mar 30, 2026

An Explicit Surrogate for Gaussian Mixture Flow Matching with Wasserstein Gap Bounds

We study training-free flow matching between two Gaussian mixture models (GMMs) using explicit velocity fields that transport one mixture into the other over time. Our baseline approach constructs component-wise Gaussian paths with affine velocity fields satisfying the continuity equation, which yields to a closed-form surrogate for the pairwise kinetic transport cost. In contrast to the exact Gaussian Wasserstein cost, which relies on matrix square-root computations, the surrogate admits a simple analytic expression derived from the kinetic energy of the induced flow. We then analyze how closely this surrogate approximates the exact cost. We prove second-order agreement in a local commuting regime and derive an explicit cubic error bound in the local commuting regime. To handle nonlocal regimes, we introduce a path-splitting strategy that localizes the covariance evolution and enables piecewise application of the bound. We finally compare the surrogate with an exact construction based on the Gaussian Wasserstein geodesic and summarize the results in a practical regime map showing when the surrogate is accurate and the exact method is preferable.

Elham Rostami, Taous-Meriem Laleg-Kirati, Hamidou Tembine

Université Paris-Saclay, Inria, CIAMS, Gif-sur-Yvette, 91190, France · Department of Electrical Engineering and Computer Science, University of Quebec in Trois-Rivieres (UQTR), Quebec, Canada

Flow MatchingOptimal TransportSurrogate ModelingWasserstein DistanceGaussian Mixture Models
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