Flow-Based Generative Modeling

Follow this topic Ask about this topic

Momentum

66 papers in the last four weeks, up 288% on the four weeks before. 0.6% of all new papers.

Jul 13Week of Sep 28

Latest papers 337

Sep 23, 2026cs.CV

On the Diffusibility of High-Dimensional Latents

Representation Autoencoders (RAEs) enable diffusion models to operate in the feature spaces of pretrained visual encoders. However, many off-the-shelf encoders are not optimized for faithful reconstruction, discarding fine-grained visual details. As expected, finetuning these encoders for image reconstruction recovers such details. However, perhaps counterintuitively, this procedure reduces the effective dimensionality of the resulting representation, and the altered geometry has downstream effects on generation. Specifically, we show that using the standard velocity prediction in flow matching in this high-dimensional space requires the model to fit orthogonal noise directions outside the low-dimensional signal manifold, making optimization inefficient. This motivates using the clean data parameterization (

\boldsymbol{x}_{0}

-prediction) instead, which focuses learning on the underlying signal manifold. Across experiments with multiple strong-reconstruction encoders, we show that

\boldsymbol{x}_{0}

-prediction consistently improves text-to-image generation performance.

Chao Feng, Zhiyang Xu, Bowei Chen +7

Cornell University · Virginia Tech · University of Washington +2

Flow Matching Latent Diffusion Models Autoencoders T2I Generation Flow-Based Generative Modeling Diffusion Models

Sep 23, 2026cs.CV

Frozen Flows Forget: Diagnosing and Restoring Lost Motion in a Latent-flow World Model

Latent world models that integrate a flow in a frozen self supervised latent space train stably and cheaply, yet silently lose the property manipulation depends on most: motion. The pretrained flow never moves the manipulated object; retraining it with latent-only losses only trades stillness for teleport-like motion. We trace the failure to the training signal, not the representation: anchor-sparse, latent-only supervision never says where along the horizon change belongs. Decode-augmented rollout training (DART) repairs this while keeping the representation frozen, retraining only the flow with decode-path supervision. DART outperforms its latent only parent on the full protocol, restores the temporal structure of motion, and re-couples predicted motion to the scene; at larger scale it further improves prediction quality, closing nearly half the remaining gap to an oracle-informed interpolation reference. Finally, we report an unexpected finding about evaluation: pixel error alone rewards frozen predictions.

Xiwen Chen, Rigaudiere Z. Li, Zhiruo Zhou +2

Shenzhen International Graduate School, Tsinghua University · Shanghai Jiaotong University · Wuhan University of Technology

World Model Learning Latent Dynamics Modeling Latent World Models Flow-Based Generative Modeling

Sep 23, 2026eess.AS

UNITE-AUDIO: Joint Learning of Continuous Tokenization and Latent Flow Matching for Text-to-Audio Generation

Text-to-audio (TTA) generation aims to synthesize realistic audio that faithfully reflects natural-language descriptions. Most TTA systems adopt a two-stage latent paradigm: an audio tokenizer is optimized for reconstruction and then frozen, after which a generative model is trained in the resulting latent space. However, reconstruction-oriented representations may be suboptimal for generation, motivating joint representation and generative learning. To this end, we introduce Unite-Audio, to our knowledge, is the first to jointly learn continuous audio representations and latent flow matching for TTA. By coupling reconstruction with self-supervised generative prediction, Unite-Audio allows the generative objective to directly shape the latent space rather than treating it as a fixed intermediate representation. We further employ Flow-GRPO post-training to improve text-conditioned generation. Experiments show competitive TTA performance with a compact latent flow model, while ablation studies confirm the benefit of jointly learning the audio representation and generative model. Audio samples are available at https://runwushi.github.io/Unite-Audio.

Runwu Shi, Kai Li, Yujin Wang +8

Institute of Science Tokyo · Tsinghua University · Wuhan University +2

Text-to-Audio Generation Audio Representation Learning Audio Tokenization Flow-Based Generative Modeling

Sep 22, 2026cs.LG

WTF?! Simulation-Free Reinforcement Learning with Wasserstein-Tilted Flow Maps

Reward fine-tuning aims to update a pre-trained flow-based generative model to improve the downstream reward of its generated samples. Existing methods typically formulate this problem as sampling from a reward-tilted distribution, the solution to a KL-regularized reward-maximization problem. Here, we introduce an optimal transport regularizer built directly from the pre-trained drift. Unlike KL reward tilting, the resulting objective transports individual samples toward higher reward rather than reweighting the base distribution. We show that the resulting problem is equivalent to a deterministic optimal control problem on the flow. Given a pre-trained flow map, this equivalence yields a simulation-free reinforcement learning algorithm for fine-tuning generative flows. We call the resulting framework Wasserstein-Tilted Flow Maps (WTF), the first end-to-end fine-tuning recipe native to flow maps. The output is a fine-tuned flow map that retains strong reward-aligned performance at few-step inference budgets without post-hoc distillation. Experiments on ImageNet-256 and text-to-image show that WTF achieves higher reward with comparable or higher diversity than baselines, while requiring up to

280\times

less training compute. More broadly, we argue that accelerated samplers such as flow maps are essential infrastructure for efficient post-training, and that the dominant KL-regularized formulation is only one of many choices worth revisiting.

Abbas Mammadov, Jerry Y. Huang, Justin Lin +5

University of Oxford · Carnegie Mellon University

Reinforcement Learning Generative Flow Networks RL for Image Generation Flow-Based Generative Modeling RL for Generative Models RL Fine-Tuning+1

Sep 22, 2026cond-mat.mtrl-sci

Topology-Stratified Materials Discovery with A Flow-Based Generative Model

Accurate generation of crystal structures is the foundation to the discovery of high-performance materials for extreme-environment applications, such as aerospace, additive manufacturing, and fusion energy systems. Although generative modeling has emerged as a promising approach for crystal design, its performance remains limited by the complex crystal structures and diverse chemical compositions. In this work, we develop UFO-MGen, a universal flow-based generative model that learns topological features of Wyckoff representations and leverages this information to accurately generate crystals across vast structural and chemical spaces. Compared with state-of-the-art generative models, UFO-MGen achieves the highest crystal generation success rate under a rigorous multi-stability evaluation framework, the highest SUN (stable, unique, novel) rate, and a remarkable extrapolation capability that has not been reported by previous models. Furthermore, a fine-tuning module is implemented to UFO-MGen for property-constrained crystal generation, enabling the inverse materials design toward target properties. The UFO-MGen opens a new avenue for accelerated materials discovery and providing a foundation for universal materials intelligence.

Jingyi Zhou, Oyshee Chowdhury, Noah Oyeniran +1

Department of Aerospace Engineering and Mechanics, University of Alabama, Tuscaloosa, 35487, Alabama, United States. · Alabama Materials Institute, The University of Alabama, Tuscaloosa, 35487, Alabama, United States.

Materials Science Generative Modeling Crystal Structure Generation Flow-Based Generative Modeling

Sep 22, 2026cs.LG

Gaussian Flow-Matching Schedules: Implications for Sampling and Training

Flow-matching schedules affect both sampling dynamics and the variance of the regression target. For centered commuting Gaussians, we show that a direction-dependent schedule decomposes into two independent design choices: a variance path, which fully determines the intermediate laws and probability flow, and a factorization, which leaves this flow unchanged while controlling irreducible regression variance. On the sampling side, we analyze finite-step Euler accuracy and derive a necessary drift bound for exact N -step sampling, connecting the geodesic and the logarithmic path. On the training side, for any fixed path, we derive closed-form factorizations that either minimize time-averaged regression variance or make it constant along the path.

Arsène Claustre, Hugo Negrel, Claire Boyer +2

DI-ENS · CNRS, ENS Paris · DMA, CFM +6

Flow Matching Diffusion Sampling Flow-Based Generative Modeling Variance Reduction

Sep 22, 2026cs.LG

When Riemann flows with Wasserstein: Generative Modeling of Probability Distributions on Manifolds

Many scientific datasets, such as molecular conformational ensembles or single-cell tissue measurements, are naturally modeled as meta-distributions: distributions over probability measures on non-Euclidean domains. Existing generative methods largely assume Euclidean geometry and fail to capture this structure. We introduce Riemannian Wasserstein Entropic Flow Matching (RWEFM), a generative framework on the Wasserstein space

\mathcal{P}_2(\mathcal{M})

of a Riemannian manifold

(\mathcal{M},g)

. RWEFM is trained by regressing a neural vector field onto Riemannian optimal transport velocities, using McCann displacement interpolations as conditional paths. We confirm theoretically that this construction leads to a valid flow matching approach on

\mathcal{P}_2(\mathcal{M})

and introduce the Riemannian Entropic Map, a GPU-efficient approximation of the optimal transport map on manifolds. Our experiments show that by respecting the intrinsic geometry of the data, RWEFM can generate whole single-cell samples in hyperspherical latent spaces and protein conformational ensembles on the torus. As RWEFM requires only a geodesic distance and a projection operator, it is not restricted to manifolds with closed-form geometry, which we demonstrate by generating distributions on a general triangulated mesh.

Doron Haviv, Edward De Brouwer, Rishabh Anand +4

Genentech Inc., South San Francisco, CA. · Yale University, New Haven, CT.

Riemannian Flow Matching Generative Modeling Flow-Based Generative Modeling Optimal Transport

Sep 21, 2026cs.LG

Corrective Forcing: Unified Post-Training for Diffusions and Flows in Generative Speech Enhancement

Diffusion and flow models, as promising generative paradigms for speech enhancement, face a training--inference mismatch: training uses analytical path states, whereas inference recursively evaluates models on self-generated rollout states along discretized sampling trajectories. This mismatch causes prediction and discretization errors to accumulate. To address it, we introduce Corrective Forcing (CoF), a post-training paradigm that forces diffusion and flow models to learn from self-generated rollouts and correct their predictions. CoF corrects clean-speech predictions on rollout states toward the ground truth under dynamic sampling schedules, exposing the model to varying inference conditions. It further regularizes local evolution using locally corrected counterfactual transitions as references for factual transitions. By expressing model outputs through a shared clean-speech prediction parameterization, CoF applies the same post-training objective across diffusion and flow formulations. Experiments with SB-VE and OT-CFM demonstrate improvements in perceptual quality and reconstruction fidelity, together with robust performance across different numbers of sampling steps.

Qing Yao, Lijian Gao, Qirong Mao

School of Computer Science and Communication Engineering, Jiangsu University Jiangsu Engineering Research Center of Big Data Ubiquitous Perception and Intelligent Agriculture Applications Provincial Key Laboratory of Computational Intelligence and New Technologies in Low-Altitude Digital Agriculture Zhenjiang, China

Diffusion Model Fine-Tuning Speech Enhancement Flow-Based Generative Modeling

Sep 21, 2026cs.CV

Classifier-Free Guidance in Flow Matching: Non-Autonomous Potentials, Overshoot, and Posterior-Mean Control

Classifier-free guidance (CFG) improves conditional generation in Flow Matching, but strong guidance can distort the generated distribution and reduce diversity. We provide a geometric account of this behavior by viewing Flow Matching as a time-varying gradient flow and characterizing how CFG reshapes its underlying potential. This view explains how stronger alignment can be accompanied by mean displacement and trajectory concentration, and motivates controlling guidance through the model-implied terminal posterior mean. We therefore propose Posterior-Mean-Capped CFG (PMC-CFG), a training-free, per-sample method that adaptively retains the strongest feasible guidance without additional network evaluations. Experiments on synthetic and large-scale image-generation benchmarks show that PMC-CFG limits guidance-induced distortion and concentration while improving the alignment--diversity trade-off, with particularly strong benefits when nominal guidance is large.

Jishen Peng, Zheng Ma

School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai, China · Institute of Natural Sciences, MOE-LSC, Shanghai Jiao Tong University, Shanghai, China · Qing Yuan Research Institute, Shanghai Jiao Tong University, Shanghai, China +1

Flow Matching Classifier-Free Guidance Conditional Image Generation Flow-Based Generative Modeling

Sep 20, 2026stat.ML

Stochastic Flow Map for Count Data

High-dimensional count data are common in scientific applications, but most diffusion and flow models are designed for continuous or categorical data, and generation often requires many sequential model evaluations. We propose Count Flow Map, a generative model that learns finite-time transitions directly in count space for one- or few-step generation. Our model directly learns stochastic transitions over finite time intervals, using Poisson births and Binomial deaths to preserve nonnegative integer counts without a predefined maximum. These transition models are trained to match the underlying local birth--death dynamics and to maintain consistency across step sizes. We characterize the connection between local dynamics and finite-time transition consistency and derive a bound on the generation error. After validating Count Flow Map in several simulations, including a high-dimensional, high-count setting, we apply it to single-cell drug perturbation prediction and neural population forecasting, where it captures perturbation responses and supports forecasts of high-activity events with only one or a few model evaluations. Together, these experiments demonstrate that Count Flow Map enables high-quality generation directly in count space across inference budgets, from one-step to few-step generation, using a single trained model.

Ganchao Wei

Flatiron Institute New York, NY 10010, USA

Synthetic Data Generation One-Step Generative Modeling Flow Map Learning Flow-Based Generative Modeling

Sep 18, 2026cs.LG

$λ$ -Controlled GRPO: Turning Flow-Matching Ratio Instability into a Budgeted Resource

Reinforcement learning is increasingly used to align image generators with reward signals, and Flow-GRPO recently extended this paradigm to flow-matching models by treating the denoising sampler as a stochastic policy that can be optimized from reward feedback. Training in this setting is unstable in a way specific to multi-step denoising: the policy update changes systematically across denoising steps, with importance ratios drifting below one, becoming increasingly dispersed, clipping at different rates, and leaving fewer usable samples late in training. Prior work treats these effects as separate failure modes and addresses each with a hand-tuned stabilizer. We show instead that they arise from a single per-step quantity, which we call path variance. This quantity is determined exactly by the sampler's Gaussian transition kernel and can be estimated cheaply during training. This reframes instability as a resource that can be measured and budgeted rather than a collection of symptoms to repair. Our method,

λ

-Controlled GRPO, calibrates importance-ratio behavior from this predicted law rather than from noisy empirical statistics, and allocates gradient effort across denoising steps according to their predicted cost. The two scales governing the update are fixed by standard policy choices rather than introduced as free tuning parameters. On a text-to-image model under two reward settings, rendering difficult target text scored by optical character recognition and matching human preferences scored by a preference model,

λ

-Controlled GRPO improves both text accuracy and preference reward over the strongest empirical stabilizer. It also keeps late-step path variance within its intended budget, precisely where the baseline systematically overshoots. The result is a Flow-GRPO update calibrated by its own transition law rather than stabilized after instability appears.

Yufeng Wang, Parivesh Priye, Meeshawn Marathe +1

Stony Brook University · Georgia Institute of Technology · Rivian and Volkswagen Group Technologies

Flow Matching T2I Generation Group Relative Policy Optimization RL for Image Generation Flow-Based Generative Modeling

Sep 17, 2026cs.CV

FlowSGS: Improving Flow Matching Priors for Inverse Imaging with Stochastic Interpolants

Flow matching has emerged as the state-of-the-art generative model and has been used for plug-and-play (PnP) priors to solve inverse problems in computational imaging. However, existing flow-based inverse solvers assume linear forward models and/or make simplifying approximations in posterior sampling. To circumvent these problems, we introduce FlowSGS, a flow-based posterior sampling method using Split Gibbs Sampling (SGS) to decompose the posterior into a likelihood step and a prior step. Specifically, we sample from the likelihood step using Langevin dynamics and leverage the Stochastic Interpolants (SI) framework to integrate a pretrained flow model into the prior step. We provide a form for the prior step that uses SI's reverse-time SDE, and show connections to previous PnP methods. Moreover, with the aid of the flow prior's straight probability paths and a novel timestep correction technique for the reverse-time SDE, FlowSGS requires fewer network evaluations in its prior step than plug-and-play diffusion samplers. Our experiments show state-of-the-art performance on a range of inverse problems. For the first time, we provide an experiment on a nonlinear inverse problem (Fourier phase retrieval) for flow-based inverse solvers.

Tianao Li, Xinhui Qian, Emma Alexander

Department of Computer Science, Northwestern University · NSF-Simons AI Institute for the Sky (SkAI) · Department of Statistics and Data Science, Northwestern University

Flow Matching Markov Chain Monte Carlo Posterior Sampling Flow-Based Generative Modeling Image Inverse Problems

Sep 16, 2026cs.LG

How to Guide Your Language Flow

We introduce a new method to guide flow matching models. Our approach, which we call probe guidance, uses the frozen internal states of an existing diffusion model to construct a guidance signal. This works using a similar principle as autoguidance, but eliminates the need for an additional forward pass at inference time and provides a reliable path to ensure that the weak and strong model share similar dynamics. We apply and benchmark this method on continuous diffusion language models, where probe guidance sets a new state-of-the-art performance on unconditional generation. When applied to a 1.7B diffusion language model, probe guidance consistently improves on multiple choice question answering benchmarks. Using our probes, we study the traditional autoguidance setting where the strong model is a weak checkpoint, and find that the weak model must come from a low-entropy region of training. These findings both provide a practical way to improve diffusion language models and shed light on the actual mechanism behind autoguidance, which is currently poorly understood.

Rohit Dilip, Tianrong Chen, Yuyang Wang +3

Apple · Caltech

Flow Matching Diffusion Model Guidance Diffusion Guidance Diffusion Language Model Inference Flow-Based Generative Modeling Diffusion Language Models+1

Sep 16, 2026cs.CV

Beyond Random Couplings: Contrastive Noise Alignment in Generative Flows

Diffusion and flow-matching models are typically trained by corrupting data through independently sampled Gaussian noise. While simple and scalable, this forward process induces arbitrary data-noise couplings, forcing the network to learn high-curvature transports between unrelated endpoints. Existing optimal-transport methods reduce this burden by reassigning fixed noise samples to data, but the source noise distribution itself remains passive. To address this, we introduce Contrastive Noise Alignment (CNA), a training-time method that creates dynamic, contrastive couplings by optimizing the noise representations directly. By modeling the noise batch as an interacting particle system, CNA employs a cross-modal InfoNCE objective to align noise particles with their paired data targets. To prevent spatial collapse, this alignment is regularized using an angular entropy term and a radial norm penalty. We show theoretically that this equilibrium asymptotically preserves Gaussian structures, maintaining tractability during inference. Empirically, CNA improves the alignment between noise and data, reduces flow curvature, and provides better generation quality with fewer required sampling steps. For few-step, pixel-space generation (2-4 NFEs), CNA reduces FID by over 50% compared to standard rectified flow, and by at least 24% against Optimal Transport baselines.

Lennart Wittke, Vinicius Azevedo

ETH Zürich · Disney Research | Studios

Flow Matching Cross-Modal Alignment Pixel-Space Diffusion Models Few-Step Diffusion Sampling Flow-Based Generative Modeling

Sep 15, 2026cs.LG

Same Flow, Different Paths: Variance Reduction in Flow Matching

In flow matching (FM), a velocity model

v_θ

is trained using a predefined path

g_t

that connects data and noise samples (e.g.,

g_t(x_0, x_1) = (1 - t) x_0 + t x_1

). In this work, we study the choice of this path from an optimization perspective by analyzing the variance of stochastic gradients. We consider the class

G(p_t,v^\star_t)

of paths that induce the same marginal distributions

p_t

and marginal velocity field

v^\star_t

, and therefore the same FM objective. Our main finding is that the choice of path

g_t

can fundamentally change the convergence rate of SGD, even when the FM objective remains exactly the same. (i) For a linear velocity model and one-dimensional Gaussian data, we derive a tight bound on the SGD iteration complexity up to logarithmic factors and find an analytically optimal path that minimizes this bound among linear paths inducing the same FM problem. (ii) We then extend the variance analysis to general FM problems and formulate path selection at a fixed

θ

as the variance-minimization problem PathOpt

_θ

, constrained to

g_t\in G(p_t,v^\star_t)

. We show that this constraint is essential: reducing variance without it can lead to slower convergence. (iii) Since the constraint

g_t \in G(p_t,v^\star_t)

cannot generally be verified directly, we derive an equivalent formulation with constraints that can be estimated from samples, allowing paths to be found numerically. Our theoretical results are supported by experiments with Gaussian data, Gaussian mixture models, and real datasets.

Alexander Tyurin

AXXX, Moscow, Russia · Applied AI Institute, Moscow, Russia

Flow Matching Stochastic Optimization Convergence Flow-Based Generative Modeling Variance Reduction Stochastic Gradient Descent

Sep 15, 2026cs.LG

FlowATC: Aircraft Trajectory Prediction via Flow Matching

Building accurate decision-support tools for next-generation air traffic control requires robust trajectory prediction models. We present a flow-matching architecture trained exclusively on historical aircraft trajectories, with no route labels or chart supervision. Trained on 1.15 million Automatic Dependent Surveillance-Broadcast trajectory windows collected over the San Francisco Bay Area, the model generates aircraft trajectory distributions that closely match historical traffic, reproducing known airspace structure around San Francisco Airport such as the shape of SFO's published NIITE FOUR departure procedure. Our model is trained directly on the native, irregular ADS-B sampling interval. Trajectory prediction is cast as sequence inpainting using a block-causal Transformer that denoises future state tokens conditioned on the observed history using Conditional Flow Matching or Denoising Diffusion Probabilistic Models. We compare our architecture against constant-velocity, deterministic-Long Short Term Memory, and Conditional Variational Autoencoders baselines. At matched parameter count, CFM outperforms DDPM by 11-26% in minADE@20, and both generative objectives surpass the CVAE baseline by 31-41%. We further show that the error degrades gracefully with prediction horizon, and the architecture remains effective when retrained on temporally decimated feeds. Lastly, we sample

K

independent completions, yielding spatial probabilistic occupancy estimates that can serve as input to downstream conflict-risk estimation.

Mathurin Petit, Emir Torun, Louis Brusset +2

École Polytechnique, Palaiseau, 91128, France · Technische Universität Berlin, Berlin, 10623, Germany · Mines Paris–PSL University, Paris, 75006, France +2

Flow Matching Air Traffic Management Flow-Based Generative Modeling Trajectory Prediction Diffusion Models

Sep 14, 2026cs.LG

Discrete Beckmann Transport Models for One-Step Language Modeling and Reasoning

Discrete diffusion and flow models are a promising alternative to autoregressive language models, but compressing many-step sampling into fewer steps typically requires distilling a pretrained teacher model. This caps the student at the teacher's quality and requires a costly two-stage training pipeline. We introduce Discrete Beckmann Transport Models (DBTM), built on a time-independent flow whose autonomous transport map provably carries any point in the ambient space to a fixed point on the vertices of the simplex in a single step. We show that this fixed-point property is characterized by a conservation equation whose residual can be minimized directly from data, removing the requirement for a teacher flow and time conditioning. Under this construction, a partially trained map corresponds to the flow truncated at finite time, so generation reduces to iterating one map until it reaches a fixed point. We further extend the map to a partial-context interpolant where additional function evaluations act as refinement steps rather than ODE integration steps. On language modeling and reasoning tasks, DBTM enables one- and few-step generation that improves quality and accuracy over discrete diffusion and continuous flow baselines.

Sophia Tang, Shiyi Wang

University of Pennsylvania · Harvard University · Kempner Institute +1

One-Step Generative Modeling Flow-Based Generative Modeling Diffusion Language Models LLM Reasoning

Sep 14, 2026cs.LG

Branched Optimal Transport Amortization

Methods of Branched Optimal Transport (BOT) mimic the economy and efficiency of natural tree-like structures, such as those found in rivers and biological systems. These methods are widely applicable for designing efficient networks in society, from river basins and blood vessels to mail and gas distribution systems. However, they remain understudied in the context of designing deep generative models, particularly at a large scale. Standard continuous-time generative models, such as the flow matching approach, fail to capture the inherent hierarchical and branching patterns present in real-world data. Current models provide no mechanism for flows to merge or share pathways to minimize total transport cost. Inspired by the "economy of scale" principle in BOT, we introduce a novel, scalable branched flow-matching algorithm designed to solve the branched optimal transport problem in high dimensions. Our method adapts the Benamou-Brenier continuous-time optimal transport formulation to learn branched generative flows. These flows allow probability mass to aggregate along common pathways before branching out to diverse targets. Parametrized by neural networks, our method effectively learns complex branched generative processes. We demonstrate its effectiveness on challenging high-dimensional tasks in biology and image generation.

Semyon Semenov, Viktor Kovalchuk, Meir Roketlishvili +4

MBZUAI

Flow Matching Generative Modeling Flow-Based Generative Modeling Optimal Transport

Sep 12, 2026cs.LG

CyFM: Cylindrical Optimal Transport for Few-Step Complex-Valued Flow Matching

Complex-valued signals like MRI and audio spectrograms are typically modelled as flat two-channel Euclidean data. The inherited Euclidean metric

dA^2 + A^2 dθ^2

vanishes at the origin, leaving phase unpenalised exactly where the signal is weakest. We replace it with the decoupled product metric

dA^2 + dθ^2

on the cylindrical closure

[0, \infty) \times S^1

, which stays non-degenerate at

A = 0

. We measure what this substitution costs and buys. Exact analytical bridges across synthetic fields, fastMRI knee data, and LibriSpeech spectrograms show Cartesian paths induce a heavy-tailed angular velocity distribution (Pareto index

\approx 1

). Under independent coupling, 43%-49% of signal energy falls on paths turning faster than

π

rad per unit time. Cylindrical paths never reach this speed. We formulate Cylindrical Flow Matching (CyFM) to strictly bound the angular regression target, coupling noise and data via exact minibatch Optimal Transport jointly over whole fields. This coupling reduces few-step generation error by 3%-60%. CyFM achieves lower generative error than the best Cartesian baseline at every step up to

k = 8

on synthetic fields and speech spectrograms, with all seeds separated. On knee MRI, the single-step advantage is 1.8x. At convergence (

k = 100

), the two geometries show no significant difference. Finally, a prior-only control exposes the cost of flat parametrisation: on synthetic fields, a single Cartesian Euler step performs worse than the unintegrated noise prior (0.376 vs. 0.150).

Marcel Musiałek, Iga Wolanin, Damian Ryczko +2

Wrocław University of Science and Technology Wrocław, Poland

Flow Matching One-Step Generative Modeling Generative Modeling Flow-Based Generative Modeling Optimal Transport

Sep 12, 2026cs.LG

Model-Aware Schedules Improve Generation via Fiberwise Optimal Transport

Diffusion and flow-matching schedules control the signal and noise coefficients that mix data and noise along affine probability paths. Minimizing a kinetic action defined on coefficient paths, motivated by optimal transport, helps explain strong baselines but remains model-agnostic and ignores prediction error. Here we introduce a model-aware schedule construction based on fiberwise optimal transport. At a fixed time and state on the probability path, compatible signal/noise decompositions form an affine fiber. We define a fiberwise prediction risk by averaging optimal-transport costs between the true and predictor-induced decompositions within these fibers. On a fixed coefficient curve, combining this risk with coefficient-path kinetic action yields a closed-form optimal time allocation. This construction extends to general linear prediction targets, and the risk profile can be estimated from an early baseline checkpoint. We evaluate DDPMs and flow matching across prediction targets, training configurations, risk-estimation checkpoints, datasets, and architectures. Our model-aware schedules consistently outperform strong baselines, including a 38.6% relative FID reduction for flow matching on CIFAR-10 at 16 function evaluations. Each model-agnostic kinetic baseline determines its own kinetic reference coordinate. In these coordinates, fiberwise-risk profiles from independently trained models in different settings align closely after normalization to unit area. The resulting schedule deformations used in training also align, suggesting empirical universality across the evaluated models and settings. Pretrained-checkpoint diagnostics extend this normalized-risk agreement to larger conditional latent diffusion and 2-RF models. A frozen analytic allocation template retains most of the model-aware improvement without further risk estimation or model-specific fitting.

Luyi Jia, Boyan Zhang, Yilun Liu +1

Arnold-Sommerfeld-Center for Theoretical Physics, Ludwig-Maximilians-Universität München, Munich, Germany · Institute of Informatics, Ludwig-Maximilians-Universität München, Munich, Germany · Munich Center for Machine Learning, Munich, Germany

Flow Matching Diffusion Timestep Scheduling Flow-Based Generative Modeling Optimal Transport Diffusion Models

Sep 11, 2026cs.LG

Flow Duality and Source Geometry for Categorical Generation

Continuous and discrete flow matching are usually treated as separate constructions. This paper identifies a duality between them: projecting continuous convex-interpolant paths with one-hot targets through a position-wise argmax yields discrete convex-interpolant paths. The result requires source laws with appropriate coordinate symmetry and boundary regularity, and it makes the continuous source distribution an explicit design choice for categorical generation. We derive the induced discrete interpolation behavior for Gaussian, bounded-uniform, and centered negative-exponential sources, showing that different source geometries lead to qualitatively different transition timing and vocabulary-size dependence. Small visual diagnostics and a short language-modeling pilot suggest that these source-design effects can also appear in learned transports and early generative quality.

Etrit Haxholli

Flow Matching Flow-Based Generative Modeling Discrete Flow Matching

Sep 11, 2026cs.LG

Particle GFlowNets: Rethinking Generative Marginalization Models

Generative Marginalization Models (MaMs) have been recently introduced as efficient neural sampling models for any-order autoregressive modelling of discrete distributions. By learning both the marginal and conditional probabilities of a persistent-block Gibbs sampler, MaMs enable fast posterior evaluation with a single neural network forward pass. While prior work has considered MaMs to be distinct from Generative Flow Networks (GFlowNets), a well-established paradigm for inference in discrete stochastic models, we show that they are equivalent. Then, we also extend MaMs' sampling strategy to non-autoregressive generative processes. In particular, we describe an automatic criterion for full-state rejuvenation of the Gibbs sampler, derived from the Gelman-Rubin statistic, which plays a key role in speeding up learning convergence. Our experiments show that our method, called Particle GFlowNets, markedly accelerates training in large combinatorial spaces.

Tiago da Silva, Diego Mesquita, Salem Lahlou

MBZUAI · School of Applied Mathematics, Getulio Vargas Foundation

Markov Chain Monte Carlo Generative Flow Networks Generative Modeling Flow-Based Generative Modeling

Sep 9, 2026stat.ML

FlowCPO: A Unified Divergence View of Preference Alignment for Flow Models

Preference alignment for flow and diffusion models now spans online reinforcement learning and offline preference optimization, but the relation between these methods remains unclear. In particular, existing forward-process alignment methods require fresh samples from the current model, while offline methods based on fixed preference pairs rely primarily on positive-only fine-tuning or DPO-style likelihood-ratio surrogates. We organize these approaches through a divergence-based framework and introduce FlowCPO, an offline forward-KL objective that uses both preferred and dispreferred samples without online rollouts. For linear interpolation, we show under explicit regularity conditions that the forward-KL objective is bounded by a contrastive flow matching loss, yielding a tractable surrogate on fixed data. We further show that this loss is nonnegative, whereas the signed regression loss of simplified FlowDPO can be unbounded below. In the in-domain setting, FlowCPO achieves higher mean GenEval and OCR scores than the evaluated baselines, reaching 0.84 and 0.87 versus 0.81 and 0.74 for FlowDPO at CFG 3.0. In the out-of-domain setting, the results are mixed, with the best GenEval result but lower reward scores than RFT on several metrics.

Yansen Han, Shengyi Liao, Peng Sun +4

Westlake University · Zhejiang University · Kling Team, Kuaishou Technology

Pairwise Preference Learning Flow Matching Flow-Based Generative Modeling Preference Alignment Offline Preference Optimization

Sep 9, 2026cond-mat.mtrl-sci

uFlowCSP: Crystal Structure Prediction using Mean flow generative models

Crystal structure prediction (CSP) is fundamental to computational materials discovery. Generative models including CDVAE, DiffCSP, FlowMM, and CrystalFlow learn stable-crystal distributions directly, but diffusion and flow-matching inference requires tens to thousands of sequential network evaluations per candidate. We introduce uFlowCSP, a MeanFlow-based CSP model that learns the average, rather than instantaneous, probability-flow velocity. It generates a complete structure in one to five evaluations, delivering 5x-58x faster inference with equal or better performance. A chemistry- and symmetry-aware Transformer uses canonical atom ordering, global composition, and per-token chemistry embeddings. A coarse crystal-system token is used only during training; it provides additive gains, particularly improving space-group agreement despite being absent at inference, which remains formula-only. On MP-20 with 20 candidates per target, one step matches CrystalFlow (78.38% vs. 78.34%) with 100x fewer evaluations and about 10x lower wall-clock time. Five steps reach 83.64%, exceeding CrystalFlow (78.34% at 2,000 evaluations) and DiffCSP (77.93% at about 20,000), while using 20x fewer evaluations. uFlowCSP generates 10,000 structures in 0.39-1.31 minutes, versus 6.5 for CrystalFlow and 76.1 for DiffCSP. Under CSPBench's energy-ranked top-five structure-and-space-group criterion, five-step uFlowCSP reaches 72%/72%/65% structure, space-group, and consensus match rates. CrystalFlow reaches 78%/73%/68% at 100 steps but falls to 49%/32%/31% at five. Thus, uFlowCSP improves accuracy per network evaluation, not merely peak accuracy.

Sourin Dey, Dipannoy Das Gupta, Lai Wei +2

Department of Computer Science and Engineering University of South Carolina Columbia, SC 29201 · Department of Computer Science and Cybersecurity University of North Georgia Columbia, SC 29201

Crystal Structure Prediction One-Step Generative Modeling Crystal Structure Generation Flow-Based Generative Modeling Conditional Generative Modeling Computational Materials Science

Sep 8, 2026stat.ML

Mode Coverage in Normalizing Flow Boltzmann Generators via Log-Ratio Variation

Normalizing flow Boltzmann generators retain a tractable pushforward density, but training with forward KL depends on target samples that may be biased or omit modes. As a result, a flow can miss target mass while its observed importance weights give a high effective sample size. We introduce the log-ratio variation

\X_ω

, the mean absolute pairwise difference of the target-to-pushforward log-density ratio under a weighting measure

ω

, and use it to define KLXX, a new loss function. Two log-ratio variations are added to the forward KL (denoted by the two X's): one weighted by the target to improve accuracy, the other by a mixture of quench and temper samples with pushforward samples to search candidate modes. We derive the Fisher--Rao gradient flow of KLXX, where both variations contribute nonpositive dissipation, and a fixed-surrogate error bound for KLXX. We use KLXX in an adaptive-staging Boltzmann generator, with importance reweighting at every stage. We bound the sampling error of its inference scheme when the stage weights are essentially bounded, and prove it asymptotically unbiased in the sample size. In the numerical tests, KLXX improves mode coverage over forward KL. It also improves the generator's per-stage diagnostics against the loss that built the schedule. The observables the generator recovers are close to independent references. The log-ratio variations thus supply information that the forward KL loss usually omits.

Qi Feng, Rongjie Lai, Di Qi +1

Department of Mathematics, Florida State University, Tallahassee, FL 32306 · Department of Mathematics, Purdue University, West Lafayette, IN 47907

Normalizing Flows Molecular Generation Flow-Based Generative Modeling Mode Collapse

Sep 8, 2026physics.chem-ph

Fixed-Dimensional Latent Flow for Generating Variable-Size 3D Molecules

Molecular size is coupled to composition, structure, and function, yet most 3D molecular generators require a predefined atom count. We introduce Equivariant-Free Transformer-Autoencoded Latent Flow Matching, a two-stage framework that samples a fixed-dimensional latent vector using flow matching and uses an autoregressive Transformer to determine molecular size, atom types, coordinates, and chemical attributes. Canonical atom ordering and rigid-pose alignment enable Transformers without equivariant layers, while decoded attributes guide bond reconstruction. On PCQM4Mv2, unconditional generation yields 87.9% unique, novel molecules passing sanitization and PoseBusters checks, exceeding baselines with lower end-to-end training and sampling time and higher end-to-end throughput. Across ten target HOMO-LUMO gaps, internal ranking retains 30% of screened candidates and increases the density functional theory-verified hit rate within 0.1 eV from 25.0% to 52.4%, while largely preserving novelty and diversity. These results demonstrate fixed-dimensional latent generation with autoregressive decoding as a practical approach to molecular design without prespecifying size.

Weichi Yao, Cameron Gruich, Bryan R. Goldsmith +1

Michigan Institute for Data & AI in Society University of Michigan, Ann Arbor, USA · Department of Chemical Engineering University of Michigan, Ann Arbor, USA · Department of Statistics University of Michigan, Ann Arbor, USA

3D Molecular Generation Molecular Generation Flow-Based Generative Modeling

Sep 3, 2026cs.CV

FlashRender: Few-Step Generative Rendering via Camera-Controlled Video MeanFlow

We present FlashRender, a few-step generative rendering framework that retakes a source video along a target camera trajectory in seconds. We identify sampling-step-dependent camera control as a prominent manifestation of discretization error in existing multi-step generative rendering models and show that resolving this inconsistency substantially lowers denoising trajectory curvature, facilitating subsequent step distillation. To this end, we introduce Representation Transformation and Alignment (RETA), which aligns hidden source-video representations with target-video features from a frozen visual geometry model. This directly encodes the geometric transformation within the source-video stream, enabling sampling-step-consistent camera control. We then fine-tune the model with the MeanFlow objective on the lower-curvature denoising trajectory induced by RETA, allowing the model to more effectively address discretization error. Finally, we apply on-policy flow map distillation to correct self-rollout errors under fixed few-step sampling. Extensive experiments show that RETA, MeanFlow, and on-policy flow map distillation play complementary roles in few-step generative rendering. Together, they enable our approach to match multi-step baselines in video quality and geometric consistency at 25x lower sampling cost while achieving superior camera controllability, even under out-of-distribution target camera trajectories.

Byeongjun Park, Byung-Hoon Kim, Hyungjin Chung

EverEx · Yonsei University · Korea University

Flow Matching Camera-Controlled Video Generation MeanFlow Neural Rendering Flow-Based Generative Modeling

Sep 3, 2026cs.LG

Beyond Straightness: Non-Crossing Flow Matching via Quantile AlignTree Coupling

The performance of Flow Matching largely depends on the quality of the coupling between the source and target distributions. However, independent coupling often leads to path crossings and local velocity ambiguity, while OT-based couplings typically incur high construction costs. To address this challenge, we propose Quantile AlignTree Flow Matching (QAT-FM), an efficient structured coupling strategy that constructs a hierarchical coupling between a Gaussian prior and the target data distribution via a quantile-aligned tree structure. QAT-FM constructs the coupling in

\mathcal{O}(Nd\log N)

time and supports per-pair source sampling with

\mathcal{O}(d)

complexity, enabling scalable training for large-scale high-dimensional generative tasks. Theoretically, we prove that the QAT coupling satisfies marginal consistency, induces non-crossing linear interpolation paths, and consistently improves path separation at intermediate times compared with independent coupling, thereby alleviating local velocity ambiguity. QAT-FM further extends naturally to conditional generation, enabling structured conditional coupling while preserving global Gaussian alignment. Experiments across diverse benchmark datasets demonstrate that QAT-FM achieves competitive generative performance while substantially reducing coupling construction cost.

Junyi Lin, Mengyu Li, Jingxuan Hu +2

Institute of Statistics and Big Data, Renmin University of China, Beijing, China · Department of Statistics and Data Science, Tsinghua University, Beijing, China

Flow Matching Flow-Based Generative Modeling

Sep 2, 2026cond-mat.stat-mech

Generative Nested Sampling of Atomistic Thermodynamic Landscapes

Nested sampling (NS) resolves the thermodynamics of an atomistic system from a single simulation, but its practical reach is limited by the Markov-chain updates needed to decorrelate walkers within each likelihood-constrained ensemble. Flow-based NS has reduced this bottleneck for gravitational-wave (GW) inference, yet its transfer to atomistic systems is not merely a change of application. Comparing a GW150914-like binary-black-hole likelihood with an eight-particle two-dimensional Lennard-Jones (LJ) system of comparable dimensionality, we show that the two landscapes differ fundamentally: atomistic multimodality is discrete and combinatorial, generated by particle permutations separated by hard collision walls, and its coordinate coupling is dense and collective, whereas the GW posterior exhibits smooth degeneracies and localized parameter coupling. Guided by this diagnosis, we introduce NS-Flows: a single conditional normalizing flow, conditioned on the NS energy bound and trained on a sliding window of recent live sets, that replaces MCMC by direct parallel draws corrected by importance-weighted rejection resampling. Live sets supply the data self-consistently, allowing flow training without structured priors or a pre-existing dataset. For LJ disks under periodic boundary conditions, the algorithm can reduce energy evaluations by two orders of magnitude and wall-clock time by about 30%, an advantage that becomes more favorable as the cost of the potential grows. The flow's generative efficiency further acts as a physical diagnostic: it varies non-monotonically along the annealing trajectory, is lowest in the dense disordered regime, and is quantitatively captured by the constrained ensemble's internal mode complexity together with target drift across the training window, identifying liquid-like ensembles, not prior-target separation, as the hard case for current flow architectures.

Alessandro Coretti, Nico Unglert, Sebastian Falkner +2

Institute of Materials Chemistry, TU Wien, 1060 Vienna, Austria · Institute of Physics, University of Augsburg, 86159 Augsburg, Germany

Normalizing Flows Flow-Based Generative Modeling Physics-Based Simulation

Sep 1, 2026cs.LG

CAT-Flow: Curvature-Adaptive sTeps for Flow Matching

Flow Matching has emerged as a leading framework for generative modeling, powering state-of-the-art systems such as FLUX and Stable Diffusion 3.5. However, the iterative nature of its ODE-based sampling process creates a fundamental efficiency bottleneck: the quality of generated samples is highly sensitive to the choice of step-sizes, and current models typically require 20 to 30 steps for good quality. In this work, we propose two lightweight, training-free algorithms, CAT-OV and CAT-OT that adapt step-sizes at inference time based on a novel connection between Flow Matching sampling and gradient flow. Our algorithms are computed efficiently by not requiring additional neural function evaluations. Specifically, CAT-OT estimates curvature over time via a finite-difference approximation of the time-derivative of the vector field, while CAT-OV approximates curvature over the state space via a gradient of the vector field. Under suitable conditions, both methods have truncation error bounds of constant order. Empirically, CAT-OV and CAT-OT outperform existing step-size heuristics in image quality metrics across four text- to-image Flow Matching models, reducing the number of generation steps required to reach comparable quality by up to 40%.

Qinchan Li, Pedro Cisneros-Velarde, Keru Fu +3

Simon Fraser University · VMware Research

Flow Matching T2I Generation Flow-Based Generative Modeling