Daily Papers

Reinforcement learning (RL) post-training improves the reward alignment of flow-based generators, but often degrades perceptual quality in ways that are not captured by the reward proxy. We identify a simple structural signature of this drift: across three post-training methods (NFT, AWM, DPO), RL fine-tuning inflates the per-step velocity norm |v_θ| by 5% to 15% relative to the reference. A form of norm inflation has been studied in classifier-free guidance (CFG), where rescaling the velocity back to a reference norm at inference time can mitigate the resulting artifacts. However, this inference-time correction does not transfer cleanly to RL: rescaling v_θ to match |v_{ref}| at inference time neither improves reward nor fixes the quality degradation, because the inflation is co-adapted into the model weights. Furthermore, an adjoint sensitivity analysis shows that velocity magnitude rescaling carries no coherent first-order reward signal at the batch level, indicating that suppressing norm inflation is unlikely to remove a consistently reward-carrying component. Since inference-time renormalization fails while norm suppression carries no reward cost, training-time intervention is the appropriate strategy. Together, these findings motivate \methodname, a hinge penalty that activates only when |v_θ| exceeds |v_{ref}| and composes additively with any velocity-local base loss. Across two base models, three post-training methods, and two reward proxies, \methodname consistently improves MLLM-judged image quality and forensic realism while preserving reward, with gains that amplify under few-step inference and are not explained by early stopping.

Generalist robot policies increasingly benefit from large-scale pretraining, but offline data alone is insufficient for robust real-world deployment. Deployed robots encounter distribution shifts, long-tail failures, task variations, and human correction opportunities that fixed demonstration datasets cannot fully capture. We present Learning While Deploying (LWD), a fleet-scale offline-to-online reinforcement learning framework for continual post-training of generalist Vision-Language-Action (VLA) policies. Starting from a pretrained VLA policy, LWD closes the loop between deployment, shared physical experience, policy improvement, and redeployment by using autonomous rollouts and human interventions collected across a robot fleet. To stabilize learning from heterogeneous, sparse-reward fleet data, LWD combines Distributional Implicit Value Learning (DIVL) for robust value estimation with Q-learning via Adjoint Matching (QAM) for policy extraction in flow-based VLA action generators. We validate LWD on a fleet of 16 dual-arm robots across eight real-world manipulation tasks, including semantic grocery restocking and 3--5 minute long-horizon tasks. A single generalist policy improves as fleet experience accumulates, reaching an average success rate of 95%, with the largest gains on long-horizon tasks.

We present MeshFlow, a new method for generating artist-like 3D meshes. Current mesh generators often adopt Auto-Regressive (AR) next-token prediction, a natural choice given the discrete nature of mesh topology. However, AR methods scale poorly because the inference cost is quadratic in mesh size. They also require discretizing the vertex coordinates, which introduces quantization errors. To address these challenges, we introduce a Variational Autoencoder (VAE) that, supervised with a contrastive loss, represents both continuous vertex positions and discrete connectivity in a continuous latent space. This latent space is significantly more compact than prior token-based mesh representations. We then build a 3D generator based on a Rectified Flow transformer, generating all mesh vertices and edges in parallel. Our model generates meshes 18x faster than the fastest AR generator while also achieving excellent accuracy across standard mesh-generation metrics. Homepage: https://mesh-flow.github.io/, Code: https://github.com/facebookresearch/meshflow

Diffusion-based imitation learning improves Behavioral Cloning (BC) on multi-modal decision-making, but comes at the cost of significantly slower inference due to the recursion in the diffusion process. It urges us to design efficient policy generators while keeping the ability to generate diverse actions. To address this challenge, we propose AdaFlow, an imitation learning framework based on flow-based generative modeling. AdaFlow represents the policy with state-conditioned ordinary differential equations (ODEs), which are known as probability flows. We reveal an intriguing connection between the conditional variance of their training loss and the discretization error of the ODEs. With this insight, we propose a variance-adaptive ODE solver that can adjust its step size in the inference stage, making AdaFlow an adaptive decision-maker, offering rapid inference without sacrificing diversity. Interestingly, it automatically reduces to a one-step generator when the action distribution is uni-modal. Our comprehensive empirical evaluation shows that AdaFlow achieves high performance with fast inference speed.

Sampling equilibrium molecular configurations from the Boltzmann distribution is a longstanding challenge. Boltzmann Generators (BGs) address this by combining exact-likelihood generative models with importance sampling, but practical scalability is limited. Meanwhile, coarse-grained surrogates enable the modeling of larger systems by reducing effective dimensionality, yet often lack a reweighting procedure required to ensure asymptotically correct statistics. In this work, we propose Coarse-Grained Boltzmann Generators (CG-BGs), a framework for reduced-order generative modeling with importance sampling in coarse-grained coordinate space. CG-BGs generate samples using a flow-based model and reweight them using a learned potential of mean force (PMF). We show that the PMF can be learned from rapidly converged trajectories via enhanced sampling force matching. Experiments demonstrate that CG-BGs capture solvent-mediated interactions in highly reduced representations while substantially reducing computational cost relative to atomistic BGs, providing a practical route toward equilibrium sampling of larger molecular systems.

Efficient sampling of molecular systems at thermodynamic equilibrium is a hallmark challenge in statistical physics. This challenge has driven the development of Boltzmann Generators (BGs), which allow rapid generation of uncorrelated equilibrium samples by combining a generative model with exact likelihoods and an importance sampling correction. However, modern BGs predominantly rely on normalizing flows (NFs), which either suffer from limited expressivity due to strict invertibility constraints (discrete time) or computationally expensive likelihoods (continuous time). In this paper, we propose Autoregressive Boltzmann Generators (ArBG) -- a novel autoregressive modelling framework -- that overcomes these limitations by departing from the flow-based BG paradigm. ArBG circumvents the topological constraints of flows and enables sequential inference-time interventions, while offering enhanced scalability by leveraging architectures effective in Large Language Models. We empirically demonstrate that ArBG leads to significant improvements over flow-based models across all benchmarks, but particularly in larger peptide systems such as the 10-residue Chignolin. Furthermore, we introduce Robin, a 132 million parameter transferable model trained with the ArBG framework which improves over the previous state-of-the-art, reducing the zero-shot energy error, E-W_2, on 8-residue systems by over 60%. The code can be found at the following link: https://github.com/danyalrehman/autobg.

Scalable sampling of molecular states in thermodynamic equilibrium is a long-standing challenge in statistical physics. Boltzmann generators tackle this problem by pairing normalizing flows with importance sampling to obtain uncorrelated samples under the target distribution. In this paper, we extend the Boltzmann generator framework with two key contributions, denoting our framework Sequential Boltzmann Generators (SBG). The first is a highly efficient Transformer-based normalizing flow operating directly on all-atom Cartesian coordinates. In contrast to the equivariant continuous flows of prior methods, we leverage exactly invertible non-equivariant architectures which are highly efficient during both sample generation and likelihood evaluation. This efficiency unlocks more sophisticated inference strategies beyond standard importance sampling. In particular, we perform inference-time scaling of flow samples using a continuous-time variant of sequential Monte Carlo, in which flow samples are transported towards the target distribution with annealed Langevin dynamics. SBG achieves state-of-the-art performance w.r.t. all metrics on peptide systems, demonstrating the first equilibrium sampling in Cartesian coordinates of tri-, tetra- and hexa-peptides that were thus far intractable for prior Boltzmann generators.

Diffusion- and flow-based models have emerged as state-of-the-art generative modeling approaches, but they require many sampling steps. Consistency models can distill these models into efficient one-step generators; however, unlike flow- and diffusion-based methods, their performance inevitably degrades when increasing the number of steps, which we show both analytically and empirically. Flow maps generalize these approaches by connecting any two noise levels in a single step and remain effective across all step counts. In this paper, we introduce two new continuous-time objectives for training flow maps, along with additional novel training techniques, generalizing existing consistency and flow matching objectives. We further demonstrate that autoguidance can improve performance, using a low-quality model for guidance during distillation, and an additional boost can be achieved by adversarial finetuning, with minimal loss in sample diversity. We extensively validate our flow map models, called Align Your Flow, on challenging image generation benchmarks and achieve state-of-the-art few-step generation performance on both ImageNet 64x64 and 512x512, using small and efficient neural networks. Finally, we show text-to-image flow map models that outperform all existing non-adversarially trained few-step samplers in text-conditioned synthesis.

Diffusion and flow-matching models have revolutionized automatic text-to-audio generation in recent times. These models are increasingly capable of generating high quality and faithful audio outputs capturing to speech and acoustic events. However, there is still much room for improvement in creative audio generation that primarily involves music and songs. Recent open lyrics-to-song models, such as, DiffRhythm, ACE-Step, and LeVo, have set an acceptable standard in automatic song generation for recreational use. However, these models lack fine-grained word-level controllability often desired by musicians in their workflows. To the best of our knowledge, our flow-matching-based JAM is the first effort toward endowing word-level timing and duration control in song generation, allowing fine-grained vocal control. To enhance the quality of generated songs to better align with human preferences, we implement aesthetic alignment through Direct Preference Optimization, which iteratively refines the model using a synthetic dataset, eliminating the need or manual data annotations. Furthermore, we aim to standardize the evaluation of such lyrics-to-song models through our public evaluation dataset JAME. We show that JAM outperforms the existing models in terms of the music-specific attributes.

Synthesizing high-fidelity and emotion-controllable talking video portraits, with audio-lip sync, vivid expression, realistic head pose, and eye blink, is an important and challenging task in recent years. Most of the existing methods suffer in achieving personalized precise emotion control or continuously interpolating between different emotions and generating diverse motion. To address these problems, we present GMTalker, a Gaussian mixture based emotional talking portraits generation framework. Specifically, we propose a Gaussian Mixture based Expression Generator (GMEG) which can construct a continuous and multi-modal latent space, achieving more flexible emotion manipulation. Furthermore, we introduce a normalizing flow based motion generator pretrained on the dataset with a wide-range motion to generate diverse motions. Finally, we propose a personalized emotion-guided head generator with an Emotion Mapping Network (EMN) which can synthesize high-fidelity and faithful emotional video portraits. Both quantitative and qualitative experiments demonstrate our method outperforms previous methods in image quality, photo-realism, emotion accuracy and motion diversity.

Normalizing flows (NFs) are end-to-end likelihood-based generative models for continuous data, and have recently regained attention with encouraging progress on image generation. Yet in the video generation domain, where spatiotemporal complexity and computational cost are substantially higher, state-of-the-art systems almost exclusively rely on diffusion-based models. In this work, we revisit this design space by presenting STARFlow-V, a normalizing flow-based video generator with substantial benefits such as end-to-end learning, robust causal prediction, and native likelihood estimation. Building upon the recently proposed STARFlow, STARFlow-V operates in the spatiotemporal latent space with a global-local architecture which restricts causal dependencies to a global latent space while preserving rich local within-frame interactions. This eases error accumulation over time, a common pitfall of standard autoregressive diffusion model generation. Additionally, we propose flow-score matching, which equips the model with a light-weight causal denoiser to improve the video generation consistency in an autoregressive fashion. To improve the sampling efficiency, STARFlow-V employs a video-aware Jacobi iteration scheme that recasts inner updates as parallelizable iterations without breaking causality. Thanks to the invertible structure, the same model can natively support text-to-video, image-to-video as well as video-to-video generation tasks. Empirically, STARFlow-V achieves strong visual fidelity and temporal consistency with practical sampling throughput relative to diffusion-based baselines. These results present the first evidence, to our knowledge, that NFs are capable of high-quality autoregressive video generation, establishing them as a promising research direction for building world models. Code and generated samples are available at https://github.com/apple/ml-starflow.

Recently, diffusion- and flow-based generative models of protein structures have emerged as a powerful tool for de novo protein design. Here, we develop Proteina, a new large-scale flow-based protein backbone generator that utilizes hierarchical fold class labels for conditioning and relies on a tailored scalable transformer architecture with up to 5x as many parameters as previous models. To meaningfully quantify performance, we introduce a new set of metrics that directly measure the distributional similarity of generated proteins with reference sets, complementing existing metrics. We further explore scaling training data to millions of synthetic protein structures and explore improved training and sampling recipes adapted to protein backbone generation. This includes fine-tuning strategies like LoRA for protein backbones, new guidance methods like classifier-free guidance and autoguidance for protein backbones, and new adjusted training objectives. Proteina achieves state-of-the-art performance on de novo protein backbone design and produces diverse and designable proteins at unprecedented length, up to 800 residues. The hierarchical conditioning offers novel control, enabling high-level secondary-structure guidance as well as low-level fold-specific generation.

Generating music that aligns with the visual content of a video has been a challenging task, as it requires a deep understanding of visual semantics and involves generating music whose melody, rhythm, and dynamics harmonize with the visual narratives. This paper presents MuVi, a novel framework that effectively addresses these challenges to enhance the cohesion and immersive experience of audio-visual content. MuVi analyzes video content through a specially designed visual adaptor to extract contextually and temporally relevant features. These features are used to generate music that not only matches the video's mood and theme but also its rhythm and pacing. We also introduce a contrastive music-visual pre-training scheme to ensure synchronization, based on the periodicity nature of music phrases. In addition, we demonstrate that our flow-matching-based music generator has in-context learning ability, allowing us to control the style and genre of the generated music. Experimental results show that MuVi demonstrates superior performance in both audio quality and temporal synchronization. The generated music video samples are available at https://muvi-v2m.github.io.

The Distribution Matching Distillation (DMD) has been successfully applied to text-to-image diffusion models such as Stable Diffusion (SD) 1.5. However, vanilla DMD suffers from convergence difficulties on large-scale flow-based text-to-image models, such as SD 3.5 and FLUX. In this paper, we first analyze the issues when applying vanilla DMD on large-scale models. Then, to overcome the scalability challenge, we propose implicit distribution alignment (IDA) to regularize the distance between the generator and fake distribution. Furthermore, we propose intra-segment guidance (ISG) to relocate the timestep importance distribution from the teacher model. With IDA alone, DMD converges for SD 3.5; employing both IDA and ISG, DMD converges for SD 3.5 and FLUX.1 dev. Along with other improvements such as scaled up discriminator models, our final model, dubbed SenseFlow, achieves superior performance in distillation for both diffusion based text-to-image models such as SDXL, and flow-matching models such as SD 3.5 Large and FLUX. The source code will be avaliable at https://github.com/XingtongGe/SenseFlow.

This paper introduces Embarrassingly Easy Text-to-Speech (E2 TTS), a fully non-autoregressive zero-shot text-to-speech system that offers human-level naturalness and state-of-the-art speaker similarity and intelligibility. In the E2 TTS framework, the text input is converted into a character sequence with filler tokens. The flow-matching-based mel spectrogram generator is then trained based on the audio infilling task. Unlike many previous works, it does not require additional components (e.g., duration model, grapheme-to-phoneme) or complex techniques (e.g., monotonic alignment search). Despite its simplicity, E2 TTS achieves state-of-the-art zero-shot TTS capabilities that are comparable to or surpass previous works, including Voicebox and NaturalSpeech 3. The simplicity of E2 TTS also allows for flexibility in the input representation. We propose several variants of E2 TTS to improve usability during inference. See https://aka.ms/e2tts/ for demo samples.

Unified multimodal models integrating visual understanding and generation face a fundamental challenge: visual generation incurs substantially higher computational costs than understanding, particularly for video. This imbalance motivates us to invert the conventional paradigm: rather than extending understanding-centric MLLMs to support generation, we propose Uni-ViGU, a framework that unifies video generation and understanding by extending a video generator as the foundation. We introduce a unified flow method that performs continuous flow matching for video and discrete flow matching for text within a single process, enabling coherent multimodal generation. We further propose a modality-driven MoE-based framework that augments Transformer blocks with lightweight layers for text generation while preserving generative priors. To repurpose generation knowledge for understanding, we design a bidirectional training mechanism with two stages: Knowledge Recall reconstructs input prompts to leverage learned text-video correspondences, while Capability Refinement fine-tunes on detailed captions to establish discriminative shared representations. Experiments demonstrate that Uni-ViGU achieves competitive performance on both video generation and understanding, validating generation-centric architectures as a scalable path toward unified multimodal intelligence. Project Page and Code: https://fr0zencrane.github.io/uni-vigu-page/.

Adversarial attacks are widely used to evaluate model robustness, and in black-box scenarios, the transferability of these attacks becomes crucial. Existing generator-based attacks have excellent generalization and transferability due to their instance-agnostic nature. However, when training generators for multi-target tasks, the success rate of transfer attacks is relatively low due to the limitations of the model's capacity. To address these challenges, we propose a novel Dual-Flow framework for multi-target instance-agnostic adversarial attacks, utilizing Cascading Distribution Shift Training to develop an adversarial velocity function. Extensive experiments demonstrate that Dual-Flow significantly improves transferability over previous multi-target generative attacks. For example, it increases the success rate from Inception-v3 to ResNet-152 by 34.58%. Furthermore, our attack method shows substantially stronger robustness against defense mechanisms, such as adversarially trained models.

Diffusion models can learn rich representations during data generation, showing potential for Self-Supervised Learning (SSL), but they face a trade-off between generative quality and discriminative performance. Their iterative sampling also incurs substantial computational and energy costs, hindering industrial and edge AI applications. To address these issues, we propose the Flow Matching-based Foundation Model (FlowFM), which jointly trains a representation encoder and a conditional flow matching generator. This decoupled design achieves both high-fidelity generation and effective recognition. By using flow matching to learn a simpler velocity field, FlowFM accelerates and stabilizes training, improving its efficiency for representation learning. Experiments on wearable sensor data show FlowFM reduces training time by 50.4\% compared to a diffusion-based approach. On downstream tasks, FlowFM surpassed the state-of-the-art SSL method (SSL-Wearables) on all five datasets while achieving up to a 51.0x inference speedup and maintaining high generative quality. The implementation code is available at https://github.com/Okita-Laboratory/jointOptimizationFlowMatching.

Current end-to-end autonomous driving systems are fundamentally limited by a mismatch between temporal causal reasoning and global trajectory consistency. Autoregressive (AR) models capture interaction-aware temporal dependencies via causal factorization, but their step-wise decoding leads to error accumulation and suboptimal global structure. In contrast, diffusion models optimize trajectories globally but lack explicit causal constraints, making them unreliable in interactive and safety-critical scenarios. This dichotomy reveals a deeper issue: existing methods treat causal modeling and global optimization as separate paradigms, without a principled way to unify them within a single trajectory distribution. To address this, we propose ChainFlow-VLA, which unifies causal generation and global refinement within a unified probabilistic framework. We formulate planning as a mixture over AR-induced modes and learn Vision-Language Model (VLM)-conditioned residual distributions over these modes. An autoregressive generator (Chain) produces a discrete set of causal trajectory modes, followed by a diffusion-based refiner (Flow) that leverages VLM hidden states as semantic priors to perform mode-conditioned correction in residual space while preserving causal structure. This straightforward conditioning seamlessly injects high-level scene understanding into fine-grained trajectory adjustments. Experiments demonstrate that ChainFlow-VLA achieves robust planning in ambiguous and long-tail scenarios, achieving a state-of-the-art score of 94.85 on the NAVSIM v1 leaderboard, matching human-level performance (94.8). Code will be available at https://github.com/AFARI-Research/ChainFlow-VLA.

Synthetic electrocardiogram generation serves medical AI applications requiring privacy-preserving data sharing and training dataset augmentation. Current diffusion-based methods achieve high generation quality but require hundreds of neural network evaluations during sampling, creating computational bottlenecks for clinical deployment. We propose FlowECG, a flow matching approach that adapts the SSSD-ECG architecture by replacing the iterative diffusion process with continuous flow dynamics. Flow matching learns direct transport paths from noise to data distributions through ordinary differential equation solving. We evaluate our method on the PTB-XL dataset using Dynamic Time Warping, Wasserstein distance, Maximum Mean Discrepancy, and spectral similarity metrics. FlowECG matches SSSD-ECG performance at 200 neural function evaluations, outperforming the baseline on three metrics. The key finding shows that FlowECG maintains generation quality with substantially fewer sampling steps, achieving comparable results with 10-25 evaluations compared to 200 for diffusion methods. This efficiency improvement reduces computational requirements by an order of magnitude while preserving physiologically realistic 12-lead ECG characteristics. The approach enables practical deployment in resource-limited clinical settings where real-time generation or large-scale synthetic data creation is needed.

Flow-based generative models are powerful exact likelihood models with efficient sampling and inference. Despite their computational efficiency, flow-based models generally have much worse density modeling performance compared to state-of-the-art autoregressive models. In this paper, we investigate and improve upon three limiting design choices employed by flow-based models in prior work: the use of uniform noise for dequantization, the use of inexpressive affine flows, and the use of purely convolutional conditioning networks in coupling layers. Based on our findings, we propose Flow++, a new flow-based model that is now the state-of-the-art non-autoregressive model for unconditional density estimation on standard image benchmarks. Our work has begun to close the significant performance gap that has so far existed between autoregressive models and flow-based models. Our implementation is available at https://github.com/aravindsrinivas/flowpp

Drag-based image editing has recently gained popularity for its interactivity and precision. However, despite the ability of text-to-image models to generate samples within a second, drag editing still lags behind due to the challenge of accurately reflecting user interaction while maintaining image content. Some existing approaches rely on computationally intensive per-image optimization or intricate guidance-based methods, requiring additional inputs such as masks for movable regions and text prompts, thereby compromising the interactivity of the editing process. We introduce InstantDrag, an optimization-free pipeline that enhances interactivity and speed, requiring only an image and a drag instruction as input. InstantDrag consists of two carefully designed networks: a drag-conditioned optical flow generator (FlowGen) and an optical flow-conditioned diffusion model (FlowDiffusion). InstantDrag learns motion dynamics for drag-based image editing in real-world video datasets by decomposing the task into motion generation and motion-conditioned image generation. We demonstrate InstantDrag's capability to perform fast, photo-realistic edits without masks or text prompts through experiments on facial video datasets and general scenes. These results highlight the efficiency of our approach in handling drag-based image editing, making it a promising solution for interactive, real-time applications.

Diffusion models have revolutionized text-to-image generation with its exceptional quality and creativity. However, its multi-step sampling process is known to be slow, often requiring tens of inference steps to obtain satisfactory results. Previous attempts to improve its sampling speed and reduce computational costs through distillation have been unsuccessful in achieving a functional one-step model. In this paper, we explore a recent method called Rectified Flow, which, thus far, has only been applied to small datasets. The core of Rectified Flow lies in its reflow procedure, which straightens the trajectories of probability flows, refines the coupling between noises and images, and facilitates the distillation process with student models. We propose a novel text-conditioned pipeline to turn Stable Diffusion (SD) into an ultra-fast one-step model, in which we find reflow plays a critical role in improving the assignment between noise and images. Leveraging our new pipeline, we create, to the best of our knowledge, the first one-step diffusion-based text-to-image generator with SD-level image quality, achieving an FID (Frechet Inception Distance) of 23.3 on MS COCO 2017-5k, surpassing the previous state-of-the-art technique, progressive distillation, by a significant margin (37.2 rightarrow 23.3 in FID). By utilizing an expanded network with 1.7B parameters, we further improve the FID to 22.4. We call our one-step models InstaFlow. On MS COCO 2014-30k, InstaFlow yields an FID of 13.1 in just 0.09 second, the best in leq 0.1 second regime, outperforming the recent StyleGAN-T (13.9 in 0.1 second). Notably, the training of InstaFlow only costs 199 A100 GPU days. Project page:~https://github.com/gnobitab/InstaFlow.

In many domains generating variable length sequences through insertions provides greater flexibility over autoregressive models. However, the action space of insertion models is much larger than that of autoregressive models (ARMs) making the learning challenging. To address this, we incorporate trainable order dynamics into the target rates for discrete flow matching, and show that with suitable choices of parameterizations, joint training of the target order dynamics and the generator is tractable without the need for numerical simulation. As the generative insertion model, we use a variable length masked diffusion model, which generates by inserting and filling mask tokens. On graph traversal tasks for which a locally optimal insertion order is known, we explore the choices of parameterization empirically and demonstrate the trade-offs between flexibility, training stability and generation quality. On de novo small molecule generation, we find that the learned order dynamics leads to an increase in the number of valid molecules generated and improved quality, when compared to uniform order dynamics.

Image inpainting techniques have shown significant improvements by using deep neural networks recently. However, most of them may either fail to reconstruct reasonable structures or restore fine-grained textures. In order to solve this problem, in this paper, we propose a two-stage model which splits the inpainting task into two parts: structure reconstruction and texture generation. In the first stage, edge-preserved smooth images are employed to train a structure reconstructor which completes the missing structures of the inputs. In the second stage, based on the reconstructed structures, a texture generator using appearance flow is designed to yield image details. Experiments on multiple publicly available datasets show the superior performance of the proposed network.

This paper introduces PeriodWave-Turbo, a high-fidelity and high-efficient waveform generation model via adversarial flow matching optimization. Recently, conditional flow matching (CFM) generative models have been successfully adopted for waveform generation tasks, leveraging a single vector field estimation objective for training. Although these models can generate high-fidelity waveform signals, they require significantly more ODE steps compared to GAN-based models, which only need a single generation step. Additionally, the generated samples often lack high-frequency information due to noisy vector field estimation, which fails to ensure high-frequency reproduction. To address this limitation, we enhance pre-trained CFM-based generative models by incorporating a fixed-step generator modification. We utilized reconstruction losses and adversarial feedback to accelerate high-fidelity waveform generation. Through adversarial flow matching optimization, it only requires 1,000 steps of fine-tuning to achieve state-of-the-art performance across various objective metrics. Moreover, we significantly reduce inference speed from 16 steps to 2 or 4 steps. Additionally, by scaling up the backbone of PeriodWave from 29M to 70M parameters for improved generalization, PeriodWave-Turbo achieves unprecedented performance, with a perceptual evaluation of speech quality (PESQ) score of 4.454 on the LibriTTS dataset. Audio samples, source code and checkpoints will be available at https://github.com/sh-lee-prml/PeriodWave.

Building on the success of diffusion models in visual generation, flow-based models reemerge as another prominent family of generative models that have achieved competitive or better performance in terms of both visual quality and inference speed. By learning the velocity field through flow-matching, flow-based models tend to produce a straighter sampling trajectory, which is advantageous during the sampling process. However, unlike diffusion models for which fast samplers are well-developed, efficient sampling of flow-based generative models has been rarely explored. In this paper, we propose a framework called FlowTurbo to accelerate the sampling of flow-based models while still enhancing the sampling quality. Our primary observation is that the velocity predictor's outputs in the flow-based models will become stable during the sampling, enabling the estimation of velocity via a lightweight velocity refiner. Additionally, we introduce several techniques including a pseudo corrector and sample-aware compilation to further reduce inference time. Since FlowTurbo does not change the multi-step sampling paradigm, it can be effectively applied for various tasks such as image editing, inpainting, etc. By integrating FlowTurbo into different flow-based models, we obtain an acceleration ratio of 53.1%sim58.3% on class-conditional generation and 29.8%sim38.5% on text-to-image generation. Notably, FlowTurbo reaches an FID of 2.12 on ImageNet with 100 (ms / img) and FID of 3.93 with 38 (ms / img), achieving the real-time image generation and establishing the new state-of-the-art. Code is available at https://github.com/shiml20/FlowTurbo.

Flow matching is a powerful framework for generating high-quality samples in various applications, especially image synthesis. However, the intensive computational demands of these models, especially during the fine-tuning process and sampling processes, pose significant challenges for low-resource scenarios. This paper introduces Bellman Optimal Step-size Straightening (BOSS) technique for distilling flow-matching generative models: it aims specifically for a few-step efficient image sampling while adhering to a computational budget constraint. First, this technique involves a dynamic programming algorithm that optimizes the step sizes of the pretrained network. Then, it refines the velocity network to match the optimal step sizes, aiming to straighten the generation paths. Extensive experimental evaluations across image generation tasks demonstrate the efficacy of BOSS in terms of both resource utilization and image quality. Our results reveal that BOSS achieves substantial gains in efficiency while maintaining competitive sample quality, effectively bridging the gap between low-resource constraints and the demanding requirements of flow-matching generative models. Our paper also fortifies the responsible development of artificial intelligence, offering a more sustainable generative model that reduces computational costs and environmental footprints. Our code can be found at https://github.com/nguyenngocbaocmt02/BOSS.

Discrete diffusion or flow models could enable faster and more controllable sequence generation than autoregressive models. We show that na\"ive linear flow matching on the simplex is insufficient toward this goal since it suffers from discontinuities in the training target and further pathologies. To overcome this, we develop Dirichlet flow matching on the simplex based on mixtures of Dirichlet distributions as probability paths. In this framework, we derive a connection between the mixtures' scores and the flow's vector field that allows for classifier and classifier-free guidance. Further, we provide distilled Dirichlet flow matching, which enables one-step sequence generation with minimal performance hits, resulting in O(L) speedups compared to autoregressive models. On complex DNA sequence generation tasks, we demonstrate superior performance compared to all baselines in distributional metrics and in achieving desired design targets for generated sequences. Finally, we show that our classifier-free guidance approach improves unconditional generation and is effective for generating DNA that satisfies design targets. Code is available at https://github.com/HannesStark/dirichlet-flow-matching.

Classifier-free guidance is a key component for enhancing the performance of conditional generative models across diverse tasks. While it has previously demonstrated remarkable improvements for the sample quality, it has only been exclusively employed for diffusion models. In this paper, we integrate classifier-free guidance into Flow Matching (FM) models, an alternative simulation-free approach that trains Continuous Normalizing Flows (CNFs) based on regressing vector fields. We explore the usage of Guided Flows for a variety of downstream applications. We show that Guided Flows significantly improves the sample quality in conditional image generation and zero-shot text-to-speech synthesis, boasting state-of-the-art performance. Notably, we are the first to apply flow models for plan generation in the offline reinforcement learning setting, showcasing a 10x speedup in computation compared to diffusion models while maintaining comparable performance.

Flow-based generative models achieve state-of-the-art sample quality, but require the expensive solution of a differential equation at inference time. Flow map models, commonly known as consistency models, encompass many recent efforts to improve inference-time efficiency by learning the solution operator of this differential equation. Yet despite their promise, these models lack a unified description that clearly explains how to learn them efficiently in practice. Here, building on the methodology proposed in Boffi et. al. (2024), we present a systematic algorithmic framework for directly learning the flow map associated with a flow or diffusion model. By exploiting a relationship between the velocity field underlying a continuous-time flow and the instantaneous rate of change of the flow map, we show how to convert any distillation scheme into a direct training algorithm via self-distillation, eliminating the need for pre-trained teachers. We introduce three algorithmic families based on different mathematical characterizations of the flow map: Eulerian, Lagrangian, and Progressive methods, which we show encompass and extend all known distillation and direct training schemes for consistency models. We find that the novel class of Lagrangian methods, which avoid both spatial derivatives and bootstrapping from small steps by design, achieve significantly more stable training and higher performance than more standard Eulerian and Progressive schemes. Our methodology unifies existing training schemes under a single common framework and reveals new design principles for accelerated generative modeling. Associated code is available at https://github.com/nmboffi/flow-maps.

Flow-based generative models (Dinh et al., 2014) are conceptually attractive due to tractability of the exact log-likelihood, tractability of exact latent-variable inference, and parallelizability of both training and synthesis. In this paper we propose Glow, a simple type of generative flow using an invertible 1x1 convolution. Using our method we demonstrate a significant improvement in log-likelihood on standard benchmarks. Perhaps most strikingly, we demonstrate that a generative model optimized towards the plain log-likelihood objective is capable of efficient realistic-looking synthesis and manipulation of large images. The code for our model is available at https://github.com/openai/glow

Generative flow networks (GFlowNets) are sequential sampling models trained to match a given distribution. GFlowNets have been successfully applied to various structured object generation tasks, sampling a diverse set of high-reward objects quickly. We propose expected flow networks (EFlowNets), which extend GFlowNets to stochastic environments. We show that EFlowNets outperform other GFlowNet formulations in stochastic tasks such as protein design. We then extend the concept of EFlowNets to adversarial environments, proposing adversarial flow networks (AFlowNets) for two-player zero-sum games. We show that AFlowNets learn to find above 80% of optimal moves in Connect-4 via self-play and outperform AlphaZero in tournaments.

Flow-based generative models parameterize probability distributions through an invertible transformation and can be trained by maximum likelihood. Invertible residual networks provide a flexible family of transformations where only Lipschitz conditions rather than strict architectural constraints are needed for enforcing invertibility. However, prior work trained invertible residual networks for density estimation by relying on biased log-density estimates whose bias increased with the network's expressiveness. We give a tractable unbiased estimate of the log density using a "Russian roulette" estimator, and reduce the memory required during training by using an alternative infinite series for the gradient. Furthermore, we improve invertible residual blocks by proposing the use of activation functions that avoid derivative saturation and generalizing the Lipschitz condition to induced mixed norms. The resulting approach, called Residual Flows, achieves state-of-the-art performance on density estimation amongst flow-based models, and outperforms networks that use coupling blocks at joint generative and discriminative modeling.

Flow based generative models have charted an impressive path across multiple visual generation tasks by adhering to a simple principle: learning velocity representations of a linear interpolant. However, we observe that training velocity solely from the final layer output underutilizes the rich inter layer representations, potentially impeding model convergence. To address this limitation, we introduce DeepFlow, a novel framework that enhances velocity representation through inter layer communication. DeepFlow partitions transformer layers into balanced branches with deep supervision and inserts a lightweight Velocity Refiner with Acceleration (VeRA) block between adjacent branches, which aligns the intermediate velocity features within transformer blocks. Powered by the improved deep supervision via the internal velocity alignment, DeepFlow converges 8 times faster on ImageNet with equivalent performance and further reduces FID by 2.6 while halving training time compared to previous flow based models without a classifier free guidance. DeepFlow also outperforms baselines in text to image generation tasks, as evidenced by evaluations on MSCOCO and zero shot GenEval.

Generating new molecules is fundamental to advancing critical applications such as drug discovery and material synthesis. Flows can generate molecules effectively by inverting the encoding process, however, existing flow models either require artifactual dequantization or specific node/edge orderings, lack desiderata such as permutation invariance, or induce discrepancy between the encoding and the decoding steps that necessitates post hoc validity correction. We circumvent these issues with novel continuous normalizing E(3)-equivariant flows, based on a system of node ODEs coupled as a graph PDE, that repeatedly reconcile locally toward globally aligned densities. Our models can be cast as message-passing temporal networks, and result in superlative performance on the tasks of density estimation and molecular generation. In particular, our generated samples achieve state-of-the-art on both the standard QM9 and ZINC250K benchmarks.

Scientific idea generation (SIG) is critical to AI-driven autonomous research, yet existing approaches are often constrained by a static retrieval-then-generation paradigm, leading to homogeneous and insufficiently divergent ideas. In this work, we propose FlowPIE, a tightly coupled retrieval-generation framework that treats literature exploration and idea generation as a co-evolving process. FlowPIE expands literature trajectories via a flow-guided Monte Carlo Tree Search (MCTS) inspired by GFlowNets, using the quality of current ideas assessed by an LLM-based generative reward model (GRM) as a supervised signal to guide adaptive retrieval and construct a diverse, high-quality initial population. Based on this population, FlowPIE models idea generation as a test-time idea evolution process, applying selection, crossover, and mutation with the isolation island paradigm and GRM-based fitness computation to incorporate cross-domain knowledge. It effectively mitigates the information cocoons arising from over-reliance on parametric knowledge and static literature. Extensive evaluations demonstrate that FlowPIE consistently produces ideas with higher novelty, feasibility and diversity compared to strong LLM-based and agent-based frameworks, while enabling reward scaling during test time.

Generating high-quality time series data has emerged as a critical research topic due to its broad utility in supporting downstream time series mining tasks. A major challenge lies in modeling the intrinsic stochasticity of temporal dynamics, as real-world sequences often exhibit random fluctuations and localized variations. While diffusion models have achieved remarkable success, their generation process is computationally inefficient, often requiring hundreds to thousands of expensive function evaluations per sample. Flow matching has emerged as a more efficient paradigm, yet its conventional ordinary differential equation (ODE)-based formulation fails to explicitly capture stochasticity, thereby limiting the fidelity of generated sequences. By contrast, stochastic differential equation (SDE) are naturally suited for modeling randomness and uncertainty. Motivated by these insights, we propose TimeFlow, a novel SDE-based flow matching framework that integrates a encoder-only architecture. Specifically, we design a component-wise decomposed velocity field to capture the multi-faceted structure of time series and augment the vanilla flow-matching optimization with an additional stochastic term to enhance representational expressiveness. TimeFlow is flexible and general, supporting both unconditional and conditional generation tasks within a unified framework. Extensive experiments across diverse datasets demonstrate that our model consistently outperforms strong baselines in generation quality, diversity, and efficiency.

Flow matching is a recent framework to train generative models that exhibits impressive empirical performance while being relatively easier to train compared with diffusion-based models. Despite its advantageous properties, prior methods still face the challenges of expensive computing and a large number of function evaluations of off-the-shelf solvers in the pixel space. Furthermore, although latent-based generative methods have shown great success in recent years, this particular model type remains underexplored in this area. In this work, we propose to apply flow matching in the latent spaces of pretrained autoencoders, which offers improved computational efficiency and scalability for high-resolution image synthesis. This enables flow-matching training on constrained computational resources while maintaining their quality and flexibility. Additionally, our work stands as a pioneering contribution in the integration of various conditions into flow matching for conditional generation tasks, including label-conditioned image generation, image inpainting, and semantic-to-image generation. Through extensive experiments, our approach demonstrates its effectiveness in both quantitative and qualitative results on various datasets, such as CelebA-HQ, FFHQ, LSUN Church & Bedroom, and ImageNet. We also provide a theoretical control of the Wasserstein-2 distance between the reconstructed latent flow distribution and true data distribution, showing it is upper-bounded by the latent flow matching objective. Our code will be available at https://github.com/VinAIResearch/LFM.git.

Taming the generation outcome of state of the art Diffusion and Flow-Matching (FM) models without having to re-train a task-specific model unlocks a powerful tool for solving inverse problems, conditional generation, and controlled generation in general. In this work we introduce D-Flow, a simple framework for controlling the generation process by differentiating through the flow, optimizing for the source (noise) point. We motivate this framework by our key observation stating that for Diffusion/FM models trained with Gaussian probability paths, differentiating through the generation process projects gradient on the data manifold, implicitly injecting the prior into the optimization process. We validate our framework on linear and non-linear controlled generation problems including: image and audio inverse problems and conditional molecule generation reaching state of the art performance across all.

Modern continuous-time generative models typically induce V-shaped flows: each sample travels independently along a nearly straight trajectory from the prior to the data. Although effective, this independent movement overlooks the hierarchical structures that exist in real-world data. To address this, we introduce Y-shaped generative flows, a framework in which samples travel together along shared pathways before branching off to target-specific endpoints. Our formulation is theoretically justified, yet remains practical, requiring only minimal modifications to standard velocity-driven models. We implement this through a scalable, neural network-based training objective. Experiments on synthetic, image, and biological datasets demonstrate that our method recovers hierarchy-aware structures, improves distributional metrics over strong flow-based baselines, and reaches targets in fewer steps.

We present Piecewise Rectified Flow (PeRFlow), a flow-based method for accelerating diffusion models. PeRFlow divides the sampling process of generative flows into several time windows and straightens the trajectories in each interval via the reflow operation, thereby approaching piecewise linear flows. PeRFlow achieves superior performance in a few-step generation. Moreover, through dedicated parameterizations, the obtained PeRFlow models show advantageous transfer ability, serving as universal plug-and-play accelerators that are compatible with various workflows based on the pre-trained diffusion models. The implementations of training and inference are fully open-sourced. https://github.com/magic-research/piecewise-rectified-flow

Flow matching (FM) is a general framework for defining probability paths via Ordinary Differential Equations (ODEs) to transform between noise and data samples. Recent approaches attempt to straighten these flow trajectories to generate high-quality samples with fewer function evaluations, typically through iterative rectification methods or optimal transport solutions. In this paper, we introduce Consistency Flow Matching (Consistency-FM), a novel FM method that explicitly enforces self-consistency in the velocity field. Consistency-FM directly defines straight flows starting from different times to the same endpoint, imposing constraints on their velocity values. Additionally, we propose a multi-segment training approach for Consistency-FM to enhance expressiveness, achieving a better trade-off between sampling quality and speed. Preliminary experiments demonstrate that our Consistency-FM significantly improves training efficiency by converging 4.4x faster than consistency models and 1.7x faster than rectified flow models while achieving better generation quality. Our code is available at: https://github.com/YangLing0818/consistency_flow_matching

Building efficient and effective generative models for neural network weights has been a research focus of significant interest that faces challenges posed by the high-dimensional weight spaces of modern neural networks and their symmetries. Several prior generative models are limited to generating partial neural network weights, particularly for larger models, such as ResNet and ViT. Those that do generate complete weights struggle with generation speed or require finetuning of the generated models. In this work, we present DeepWeightFlow, a Flow Matching model that operates directly in weight space to generate diverse and high-accuracy neural network weights for a variety of architectures, neural network sizes, and data modalities. The neural networks generated by DeepWeightFlow do not require fine-tuning to perform well and can scale to large networks. We apply Git Re-Basin and TransFusion for neural network canonicalization in the context of generative weight models to account for the impact of neural network permutation symmetries and to improve generation efficiency for larger model sizes. The generated networks excel at transfer learning, and ensembles of hundreds of neural networks can be generated in minutes, far exceeding the efficiency of diffusion-based methods. DeepWeightFlow models pave the way for more efficient and scalable generation of diverse sets of neural networks.

Generative Flow Networks (GFlowNets) are amortized sampling methods that learn a distribution over discrete objects proportional to their rewards. GFlowNets exhibit a remarkable ability to generate diverse samples, yet occasionally struggle to consistently produce samples with high rewards due to over-exploration on wide sample space. This paper proposes to train GFlowNets with local search, which focuses on exploiting high-rewarded sample space to resolve this issue. Our main idea is to explore the local neighborhood via backtracking and reconstruction guided by backward and forward policies, respectively. This allows biasing the samples toward high-reward solutions, which is not possible for a typical GFlowNet solution generation scheme, which uses the forward policy to generate the solution from scratch. Extensive experiments demonstrate a remarkable performance improvement in several biochemical tasks. Source code is available: https://github.com/dbsxodud-11/ls_gfn.

The multi-step denoising process in diffusion and Flow Matching models causes major efficiency issues, which motivates research on few-step generation. We present Solution Flow Models (SoFlow), a framework for one-step generation from scratch. By analyzing the relationship between the velocity function and the solution function of the velocity ordinary differential equation (ODE), we propose a Flow Matching loss and a solution consistency loss to train our models. The Flow Matching loss allows our models to provide estimated velocity fields for Classifier-Free Guidance (CFG) during training, which improves generation performance. Notably, our consistency loss does not require the calculation of the Jacobian-vector product (JVP), a common requirement in recent works that is not well-optimized in deep learning frameworks like PyTorch. Experimental results indicate that, when trained from scratch using the same Diffusion Transformer (DiT) architecture and an equal number of training epochs, our models achieve better FID-50K scores than MeanFlow models on the ImageNet 256x256 dataset.

Given an unconditional generative model and a predictor for a target property (e.g., a classifier), the goal of training-free guidance is to generate samples with desirable target properties without additional training. As a highly efficient technique for steering generative models toward flexible outcomes, training-free guidance has gained increasing attention in diffusion models. However, existing methods only handle data in continuous spaces, while many scientific applications involve both continuous and discrete data (referred to as multimodality). Another emerging trend is the growing use of the simple and general flow matching framework in building generative foundation models, where guided generation remains under-explored. To address this, we introduce TFG-Flow, a novel training-free guidance method for multimodal generative flow. TFG-Flow addresses the curse-of-dimensionality while maintaining the property of unbiased sampling in guiding discrete variables. We validate TFG-Flow on four molecular design tasks and show that TFG-Flow has great potential in drug design by generating molecules with desired properties.

Generative modeling of single-cell RNA-seq data has shown invaluable potential in community-driven tasks such as trajectory inference, batch effect removal and gene expression generation. However, most recent deep models generating synthetic single cells from noise operate on pre-processed continuous gene expression approximations, ignoring the inherently discrete and over-dispersed nature of single-cell data, which limits downstream applications and hinders the incorporation of robust noise models. Moreover, crucial aspects of deep-learning-based synthetic single-cell generation remain underexplored, such as controllable multi-modal and multi-label generation and its role in the performance enhancement of downstream tasks. This work presents Cell Flow for Generation (CFGen), a flow-based conditional generative model for multi-modal single-cell counts, which explicitly accounts for the discrete nature of the data. Our results suggest improved recovery of crucial biological data characteristics while accounting for novel generative tasks such as conditioning on multiple attributes and boosting rare cell type classification via data augmentation. By showcasing CFGen on a diverse set of biological datasets and settings, we provide evidence of its value to the fields of computational biology and deep generative models.

Normalising Flows are non-parametric statistical models characterised by their dual capabilities of density estimation and generation. This duality requires an inherently invertible architecture. However, the requirement of invertibility imposes constraints on their expressiveness, necessitating a large number of parameters and innovative architectural designs to achieve good results. Whilst flow-based models predominantly rely on neural-network-based transformations for expressive designs, alternative transformation methods have received limited attention. In this work, we present Ferumal flow, a novel kernelised normalising flow paradigm that integrates kernels into the framework. Our results demonstrate that a kernelised flow can yield competitive or superior results compared to neural network-based flows whilst maintaining parameter efficiency. Kernelised flows excel especially in the low-data regime, enabling flexible non-parametric density estimation in applications with sparse data availability.

Recent advances in artificial intelligence (AI) have produced highly capable and controllable systems. This creates unprecedented opportunities for structured reasoning as well as collaboration among multiple AI systems and humans. To fully realize this potential, it is essential to develop a principled way of designing and studying such structured interactions. For this purpose, we introduce the conceptual framework of Flows: a systematic approach to modeling complex interactions. Flows are self-contained building blocks of computation, with an isolated state, communicating through a standardized message-based interface. This modular design allows Flows to be recursively composed into arbitrarily nested interactions, with a substantial reduction of complexity. Crucially, any interaction can be implemented using this framework, including prior work on AI--AI and human--AI interactions, prompt engineering schemes, and tool augmentation. We demonstrate the potential of Flows on the task of competitive coding, a challenging task on which even GPT-4 struggles. Our results suggest that structured reasoning and collaboration substantially improve generalization, with AI-only Flows adding +21 and human--AI Flows adding +54 absolute points in terms of solve rate. To support rapid and rigorous research, we introduce the aiFlows library. The library comes with a repository of Flows that can be easily used, extended, and composed into novel, more complex Flows. The aiFlows library is available at https://github.com/epfl-dlab/aiflows. Data and Flows for reproducing our experiments are available at https://github.com/epfl-dlab/cc_flows.

In this paper, we establish a connection between the parameterization of flow-based and energy-based generative models, and present a new flow-based modeling approach called energy-based normalizing flow (EBFlow). We demonstrate that by optimizing EBFlow with score-matching objectives, the computation of Jacobian determinants for linear transformations can be entirely bypassed. This feature enables the use of arbitrary linear layers in the construction of flow-based models without increasing the computational time complexity of each training iteration from O(D^2L) to O(D^3L) for an L-layered model that accepts D-dimensional inputs. This makes the training of EBFlow more efficient than the commonly-adopted maximum likelihood training method. In addition to the reduction in runtime, we enhance the training stability and empirical performance of EBFlow through a number of techniques developed based on our analysis of the score-matching methods. The experimental results demonstrate that our approach achieves a significant speedup compared to maximum likelihood estimation while outperforming prior methods with a noticeable margin in terms of negative log-likelihood (NLL).

Despite Flow Matching and diffusion models having emerged as powerful generative paradigms for continuous variables such as images and videos, their application to high-dimensional discrete data, such as language, is still limited. In this work, we present Discrete Flow Matching, a novel discrete flow paradigm designed specifically for generating discrete data. Discrete Flow Matching offers several key contributions: (i) it works with a general family of probability paths interpolating between source and target distributions; (ii) it allows for a generic formula for sampling from these probability paths using learned posteriors such as the probability denoiser (x-prediction) and noise-prediction (epsilon-prediction); (iii) practically, focusing on specific probability paths defined with different schedulers considerably improves generative perplexity compared to previous discrete diffusion and flow models; and (iv) by scaling Discrete Flow Matching models up to 1.7B parameters, we reach 6.7% Pass@1 and 13.4% Pass@10 on HumanEval and 6.7% Pass@1 and 20.6% Pass@10 on 1-shot MBPP coding benchmarks. Our approach is capable of generating high-quality discrete data in a non-autoregressive fashion, significantly closing the gap between autoregressive models and discrete flow models.

Flow-based generative models have become a strong framework for high-quality generative modeling, yet pretrained models are rarely used in their vanilla conditional form: conditional samples without guidance often appear diffuse and lack fine-grained detail due to the smoothing effects of neural networks. Existing guidance techniques such as classifier-free guidance (CFG) improve fidelity but double the inference cost and typically reduce sample diversity. We introduce Momentum Guidance (MG), a new dimension of guidance that leverages the ODE trajectory itself. MG extrapolates the current velocity using an exponential moving average of past velocities and preserves the standard one-evaluation-per-step cost. It matches the effect of standard guidance without extra computation and can further improve quality when combined with CFG. Experiments demonstrate MG's effectiveness across benchmarks. Specifically, on ImageNet-256, MG achieves average improvements in FID of 36.68% without CFG and 25.52% with CFG across various sampling settings, attaining an FID of 1.597 at 64 sampling steps. Evaluations on large flow-based models like Stable Diffusion 3 and FLUX.1-dev further confirm consistent quality enhancements across standard metrics.

Generative machine learning methods, such as diffusion models and flow matching, have shown great potential in modeling complex system behaviors and building efficient surrogate models. However, these methods typically learn the underlying physics implicitly from data. We propose Physics-Based Flow Matching (PBFM), a novel generative framework that explicitly embeds physical constraints, both PDE residuals and algebraic relations, into the flow matching objective. We also introduce temporal unrolling at training time that improves the accuracy of the final, noise-free sample prediction. Our method jointly minimizes the flow matching loss and the physics-based residual loss without requiring hyperparameter tuning of their relative weights. Additionally, we analyze the role of the minimum noise level, sigma_{min}, in the context of physical constraints and evaluate a stochastic sampling strategy that helps to reduce physical residuals. Through extensive benchmarks on three representative PDE problems, we show that our approach yields up to an 8times more accurate physical residuals compared to FM, while clearly outperforming existing algorithms in terms of distributional accuracy. PBFM thus provides a principled and efficient framework for surrogate modeling, uncertainty quantification, and accelerated simulation in physics and engineering applications.

Diffusion models and flow-based methods have shown impressive generative capability, especially for images, but their sampling is expensive because it requires many iterative updates. We introduce W-Flow, a framework for training a generator that transforms samples from a simple reference distribution into samples from a target data distribution in a single step. This is achieved in two steps: we first define an evolution from the reference distribution to the target distribution through a Wasserstein gradient flow that minimizes an energy functional; second, we train a static neural generator to compress this evolution into one-step generation. We instantiate the energy functional with the Sinkhorn divergence, which yields an efficient optimal-transport-based update rule that captures global distributional discrepancy and improves coverage of the target distribution. We further prove that the finite-sample training dynamics converge to the continuous-time distributional dynamics under suitable assumptions. Empirically, W-Flow sets a new state of the art for one-step ImageNet 256times256 generation, achieving 1.29 FID, with improved mode coverage and domain transfer. Compared to multi-step diffusion models with similar FID scores, our method yields approximately 100times faster sampling. These results show that Wasserstein gradient flows provide a principled and effective foundation for fast and high-fidelity generative modeling.

Flow matching has emerged as a promising framework for training generative models, demonstrating impressive empirical performance while offering relative ease of training compared to diffusion-based models. However, this method still requires numerous function evaluations in the sampling process. To address these limitations, we introduce a self-corrected flow distillation method that effectively integrates consistency models and adversarial training within the flow-matching framework. This work is a pioneer in achieving consistent generation quality in both few-step and one-step sampling. Our extensive experiments validate the effectiveness of our method, yielding superior results both quantitatively and qualitatively on CelebA-HQ and zero-shot benchmarks on the COCO dataset. Our implementation is released at https://github.com/VinAIResearch/SCFlow

We present adversarial flow models, a class of generative models that unifies adversarial models and flow models. Our method supports native one-step or multi-step generation and is trained using the adversarial objective. Unlike traditional GANs, where the generator learns an arbitrary transport plan between the noise and the data distributions, our generator learns a deterministic noise-to-data mapping, which is the same optimal transport as in flow-matching models. This significantly stabilizes adversarial training. Also, unlike consistency-based methods, our model directly learns one-step or few-step generation without needing to learn the intermediate timesteps of the probability flow for propagation. This saves model capacity, reduces training iterations, and avoids error accumulation. Under the same 1NFE setting on ImageNet-256px, our B/2 model approaches the performance of consistency-based XL/2 models, while our XL/2 model creates a new best FID of 2.38. We additionally show the possibility of end-to-end training of 56-layer and 112-layer models through depth repetition without any intermediate supervision, and achieve FIDs of 2.08 and 1.94 using a single forward pass, surpassing their 2NFE and 4NFE counterparts.

Existing approaches to controllable generation typically rely on fine-tuning, auxiliary networks, or test-time search. We show that flow matching admits a different control interface: adaptation through examples. For deterministic interpolants, the velocity field is solely governed by a conditional endpoint mean; shifting this mean shifts the flow itself. This yields a simple principle for controllable generation: steer a pretrained model by changing the reference set it follows. We instantiate this idea in two forms. Reference-Mean Guidance is training-free: it computes a closed-form endpoint-mean correction from a reference bank and applies it to a frozen FLUX.2-klein (4B) model, enabling control of color, identity, style, and structure while keeping the prompt, seed, and weights fixed. Semi-Parametric Guidance amortizes the same idea through an explicit mean anchor and learned residual refiner, matching unconditional DiT-B/4 quality on AFHQv2 while allowing the reference set to be swapped at inference time. These results point to a broader direction: generative models that adapt through data, not parameter updates.

The ability to learn multi-modal action distributions is indispensable for robotic manipulation policies to perform precise and robust control. Flow-based generative models have recently emerged as a promising solution to learning distributions of actions, offering one-step action generation and thus achieving much higher sampling efficiency compared to diffusion-based methods. However, existing flow-based policies suffer from representation collapse, the inability to distinguish similar visual representations, leading to failures in precise manipulation tasks. We propose DM1 (MeanFlow with Dispersive Regularization for One-Step Robotic Manipulation), a novel flow matching framework that integrates dispersive regularization into MeanFlow to prevent collapse while maintaining one-step efficiency. DM1 employs multiple dispersive regularization variants across different intermediate embedding layers, encouraging diverse representations across training batches without introducing additional network modules or specialized training procedures. Experiments on RoboMimic benchmarks show that DM1 achieves 20-40 times faster inference (0.07s vs. 2-3.5s) and improves success rates by 10-20 percentage points, with the Lift task reaching 99% success over 85% of the baseline. Real-robot deployment on a Franka Panda further validates that DM1 transfers effectively from simulation to the physical world. To the best of our knowledge, this is the first work to leverage representation regularization to enable flow-based policies to achieve strong performance in robotic manipulation, establishing a simple yet powerful approach for efficient and robust manipulation.

Normalizing flows are a promising tool for modeling probability distributions in physical systems. While state-of-the-art flows accurately approximate distributions and energies, applications in physics additionally require smooth energies to compute forces and higher-order derivatives. Furthermore, such densities are often defined on non-trivial topologies. A recent example are Boltzmann Generators for generating 3D-structures of peptides and small proteins. These generative models leverage the space of internal coordinates (dihedrals, angles, and bonds), which is a product of hypertori and compact intervals. In this work, we introduce a class of smooth mixture transformations working on both compact intervals and hypertori. Mixture transformations employ root-finding methods to invert them in practice, which has so far prevented bi-directional flow training. To this end, we show that parameter gradients and forces of such inverses can be computed from forward evaluations via the inverse function theorem. We demonstrate two advantages of such smooth flows: they allow training by force matching to simulation data and can be used as potentials in molecular dynamics simulations.

Normalizing Flows (NFs) have been established as a principled framework for generative modeling. Standard NFs consist of a forward process and a reverse process: the forward process maps data to noise, while the reverse process generates samples by inverting it. Typical NF forward transformations are constrained by explicit invertibility, ensuring that the reverse process can serve as their exact analytic inverse. Recent developments in TARFlow and its variants have revitalized NF methods by combining Transformers and autoregressive flows, but have also exposed causal decoding as a major bottleneck. In this work, we introduce Bidirectional Normalizing Flow (BiFlow), a framework that removes the need for an exact analytic inverse. BiFlow learns a reverse model that approximates the underlying noise-to-data inverse mapping, enabling more flexible loss functions and architectures. Experiments on ImageNet demonstrate that BiFlow, compared to its causal decoding counterpart, improves generation quality while accelerating sampling by up to two orders of magnitude. BiFlow yields state-of-the-art results among NF-based methods and competitive performance among single-evaluation ("1-NFE") methods. Following recent encouraging progress on NFs, we hope our work will draw further attention to this classical paradigm.

Continuous-time Consistency Models (CMs) promise efficient few-step generation but face significant challenges with training instability. We argue this instability stems from a fundamental conflict: by training a network to learn only a shortcut across a probability flow, the model loses its grasp on the instantaneous velocity field that defines the flow. Our solution is to explicitly anchor the model in the underlying flow during training. We introduce the Flow-Anchored Consistency Model (FACM), a simple but effective training strategy that uses a Flow Matching (FM) task as an anchor for the primary CM shortcut objective. This Flow-Anchoring approach requires no architectural modifications and is broadly compatible with standard model architectures. By distilling a pre-trained LightningDiT model, our method achieves a state-of-the-art FID of 1.32 with two steps (NFE=2) and 1.76 with just one step (NFE=1) on ImageNet 256x256, significantly outperforming previous methods. This provides a general and effective recipe for building high-performance, few-step generative models. Our code and pretrained models: https://github.com/ali-vilab/FACM.

This paper focuses on a novel approach for detecting moving objects during camera motion. We present an optical-flow-based transformation that yields a consistent 2D invariant image output regardless of time instants, range of points in 3D, and the speed of the camera. In other words, this transformation generates a lookup image that remains invariant despite the changing projection of the 3D scene and camera motion. In the new domain, projections of 3D points that deviate from the values of the predefined lookup image can be clearly identified as moving relative to the stationary 3D environment, making them seamlessly detectable. The method does not require prior knowledge of the direction of motion or speed of the camera, nor does it necessitate 3D point range information. It is well-suited for real-time parallel processing, rendering it highly practical for implementation. We have validated the effectiveness of the new domain through simulations and experiments, demonstrating its robustness in scenarios involving rectilinear camera motion, both in simulations and with real-world data. This approach introduces new ways for moving objects detection during camera motion, and also lays the foundation for future research in the context of moving object detection during six-degrees-of-freedom camera motion.

Flow matching casts sample generation as learning a continuous-time velocity field that transports noise to data. Existing flow matching networks typically predict each point's velocity independently, considering only its location and time along its flow trajectory, and ignoring neighboring points. However, this pointwise approach may overlook correlations between points along the generation trajectory that could enhance velocity predictions, thereby improving downstream generation quality. To address this, we propose Graph Flow Matching (GFM), a lightweight enhancement that decomposes the learned velocity into a reaction term -- any standard flow matching network -- and a diffusion term that aggregates neighbor information via a graph neural module. This reaction-diffusion formulation retains the scalability of deep flow models while enriching velocity predictions with local context, all at minimal additional computational cost. Operating in the latent space of a pretrained variational autoencoder, GFM consistently improves Fr\'echet Inception Distance (FID) and recall across five image generation benchmarks (LSUN Church, LSUN Bedroom, FFHQ, AFHQ-Cat, and CelebA-HQ at 256times256), demonstrating its effectiveness as a modular enhancement to existing flow matching architectures.

We present PitchFlower, a flow-based neural audio codec with explicit pitch controllability. Our approach enforces disentanglement through a simple perturbation: during training, F0 contours are flattened and randomly shifted, while the true F0 is provided as conditioning. A vector-quantization bottleneck prevents pitch recovery, and a flow-based decoder generates high quality audio. Experiments show that PitchFlower achieves more accurate pitch control than WORLD at much higher audio quality, and outperforms SiFiGAN in controllability while maintaining comparable quality. Beyond pitch, this framework provides a simple and extensible path toward disentangling other speech attributes.

Diffusion or flow-based models are powerful generative paradigms that are notoriously hard to sample as samples are defined as solutions to high-dimensional Ordinary or Stochastic Differential Equations (ODEs/SDEs) which require a large Number of Function Evaluations (NFE) to approximate well. Existing methods to alleviate the costly sampling process include model distillation and designing dedicated ODE solvers. However, distillation is costly to train and sometimes can deteriorate quality, while dedicated solvers still require relatively large NFE to produce high quality samples. In this paper we introduce "Bespoke solvers", a novel framework for constructing custom ODE solvers tailored to the ODE of a given pre-trained flow model. Our approach optimizes an order consistent and parameter-efficient solver (e.g., with 80 learnable parameters), is trained for roughly 1% of the GPU time required for training the pre-trained model, and significantly improves approximation and generation quality compared to dedicated solvers. For example, a Bespoke solver for a CIFAR10 model produces samples with Fr\'echet Inception Distance (FID) of 2.73 with 10 NFE, and gets to 1% of the Ground Truth (GT) FID (2.59) for this model with only 20 NFE. On the more challenging ImageNet-64times64, Bespoke samples at 2.2 FID with 10 NFE, and gets within 2% of GT FID (1.71) with 20 NFE.

Few-step diffusion or flow-based generative models typically distill a velocity-predicting teacher into a student that predicts a shortcut towards denoised data. This format mismatch has led to complex distillation procedures that often suffer from a quality-diversity trade-off. To address this, we propose policy-based flow models (pi-Flow). pi-Flow modifies the output layer of a student flow model to predict a network-free policy at one timestep. The policy then produces dynamic flow velocities at future substeps with negligible overhead, enabling fast and accurate ODE integration on these substeps without extra network evaluations. To match the policy's ODE trajectory to the teacher's, we introduce a novel imitation distillation approach, which matches the policy's velocity to the teacher's along the policy's trajectory using a standard ell_2 flow matching loss. By simply mimicking the teacher's behavior, pi-Flow enables stable and scalable training and avoids the quality-diversity trade-off. On ImageNet 256^2, it attains a 1-NFE FID of 2.85, outperforming MeanFlow of the same DiT architecture. On FLUX.1-12B and Qwen-Image-20B at 4 NFEs, pi-Flow achieves substantially better diversity than state-of-the-art few-step methods, while maintaining teacher-level quality.

Diffusion and flow-based models have become the state of the art for generative AI across a wide range of data modalities, including images, videos, shapes, molecules, music, and more. This tutorial provides a self-contained introduction to diffusion and flow-based generative models from first principles. We systematically develop the necessary mathematical background in ordinary and stochastic differential equations and derive the core algorithms of flow matching and denoising diffusion models. We then provide a step-by-step guide to building image and video generators, including training methods, guidance, and architectural design. This course is ideal for machine learning researchers who want to develop a principled understanding of the theory and practice of generative AI.

We introduce a new paradigm for generative modeling built on Continuous Normalizing Flows (CNFs), allowing us to train CNFs at unprecedented scale. Specifically, we present the notion of Flow Matching (FM), a simulation-free approach for training CNFs based on regressing vector fields of fixed conditional probability paths. Flow Matching is compatible with a general family of Gaussian probability paths for transforming between noise and data samples -- which subsumes existing diffusion paths as specific instances. Interestingly, we find that employing FM with diffusion paths results in a more robust and stable alternative for training diffusion models. Furthermore, Flow Matching opens the door to training CNFs with other, non-diffusion probability paths. An instance of particular interest is using Optimal Transport (OT) displacement interpolation to define the conditional probability paths. These paths are more efficient than diffusion paths, provide faster training and sampling, and result in better generalization. Training CNFs using Flow Matching on ImageNet leads to consistently better performance than alternative diffusion-based methods in terms of both likelihood and sample quality, and allows fast and reliable sample generation using off-the-shelf numerical ODE solvers.

Finding a suitable layout represents a crucial task for diverse applications in graphic design. Motivated by simpler and smoother sampling trajectories, we explore the use of Flow Matching as an alternative to current diffusion-based layout generation models. Specifically, we propose LayoutFlow, an efficient flow-based model capable of generating high-quality layouts. Instead of progressively denoising the elements of a noisy layout, our method learns to gradually move, or flow, the elements of an initial sample until it reaches its final prediction. In addition, we employ a conditioning scheme that allows us to handle various generation tasks with varying degrees of conditioning with a single model. Empirically, LayoutFlow performs on par with state-of-the-art models while being significantly faster.

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

Flow matching models have emerged as a powerful method for generative modeling on domains like images or videos, and even on irregular or unstructured data like 3D point clouds or even protein structures. These models are commonly trained in two stages: first, a data compressor is trained, and in a subsequent training stage a flow matching generative model is trained in the latent space of the data compressor. This two-stage paradigm sets obstacles for unifying models across data domains, as hand-crafted compressors architectures are used for different data modalities. To this end, we introduce INRFlow, a domain-agnostic approach to learn flow matching transformers directly in ambient space. Drawing inspiration from INRs, we introduce a conditionally independent point-wise training objective that enables INRFlow to make predictions continuously in coordinate space. Our empirical results demonstrate that INRFlow effectively handles different data modalities such as images, 3D point clouds and protein structure data, achieving strong performance in different domains and outperforming comparable approaches. INRFlow is a promising step towards domain-agnostic flow matching generative models that can be trivially adopted in different data domains.

Diffusion models have greatly improved visual generation but are hindered by slow generation speed due to the computationally intensive nature of solving generative ODEs. Rectified flow, a widely recognized solution, improves generation speed by straightening the ODE path. Its key components include: 1) using the diffusion form of flow-matching, 2) employing boldsymbol v-prediction, and 3) performing rectification (a.k.a. reflow). In this paper, we argue that the success of rectification primarily lies in using a pretrained diffusion model to obtain matched pairs of noise and samples, followed by retraining with these matched noise-sample pairs. Based on this, components 1) and 2) are unnecessary. Furthermore, we highlight that straightness is not an essential training target for rectification; rather, it is a specific case of flow-matching models. The more critical training target is to achieve a first-order approximate ODE path, which is inherently curved for models like DDPM and Sub-VP. Building on this insight, we propose Rectified Diffusion, which generalizes the design space and application scope of rectification to encompass the broader category of diffusion models, rather than being restricted to flow-matching models. We validate our method on Stable Diffusion v1-5 and Stable Diffusion XL. Our method not only greatly simplifies the training procedure of rectified flow-based previous works (e.g., InstaFlow) but also achieves superior performance with even lower training cost. Our code is available at https://github.com/G-U-N/Rectified-Diffusion.

This paper studies generative flow networks (GFlowNets) to sample objects from the Boltzmann energy distribution via a sequence of actions. In particular, we focus on improving GFlowNet with partial inference: training flow functions with the evaluation of the intermediate states or transitions. To this end, the recently developed forward-looking GFlowNet reparameterizes the flow functions based on evaluating the energy of intermediate states. However, such an evaluation of intermediate energies may (i) be too expensive or impossible to evaluate and (ii) even provide misleading training signals under large energy fluctuations along the sequence of actions. To resolve this issue, we propose learning energy decompositions for GFlowNets (LED-GFN). Our main idea is to (i) decompose the energy of an object into learnable potential functions defined on state transitions and (ii) reparameterize the flow functions using the potential functions. In particular, to produce informative local credits, we propose to regularize the potential to change smoothly over the sequence of actions. It is also noteworthy that training GFlowNet with our learned potential can preserve the optimal policy. We empirically verify the superiority of LED-GFN in five problems including the generation of unstructured and maximum independent sets, molecular graphs, and RNA sequences.

In this work, we introduce an efficient generation procedure to produce synthetic multi-modal datasets of fluid simulations. The procedure can reproduce the dynamics of fluid flows and allows for exploring and learning various properties of their complex behavior, from distinct perspectives and modalities. We employ our framework to generate a set of thoughtfully designed training datasets, which attempt to span specific fluid simulation scenarios in a meaningful way. The properties of our contributions are demonstrated by evaluating recently published algorithms for the neural fluid simulation and fluid inverse rendering tasks using our benchmark datasets. Our contribution aims to fulfill the community's need for standardized training data, fostering more reproducibile and robust research.

Generative flow networks (GFlowNets) are a family of algorithms that learn a generative policy to sample discrete objects x with non-negative reward R(x). Learning objectives guarantee the GFlowNet samples x from the target distribution p^*(x) propto R(x) when loss is globally minimized over all states or trajectories, but it is unclear how well they perform with practical limits on training resources. We introduce an efficient evaluation strategy to compare the learned sampling distribution to the target reward distribution. As flows can be underdetermined given training data, we clarify the importance of learned flows to generalization and matching p^*(x) in practice. We investigate how to learn better flows, and propose (i) prioritized replay training of high-reward x, (ii) relative edge flow policy parametrization, and (iii) a novel guided trajectory balance objective, and show how it can solve a substructure credit assignment problem. We substantially improve sample efficiency on biochemical design tasks.

Flow-based generative models have recently demonstrated strong performance, yet sampling typically relies on expensive numerical integration of ordinary differential equations (ODEs). Rectified Flow enables one-step sampling by learning nearly straight probability paths, but achieving such straightness requires multiple computationally intensive reflow iterations. MeanFlow achieves one-step generation by directly modeling the average velocity over time; however, when trained on highly curved flows, it suffers from slow convergence and noisy supervision. To address these limitations, we propose Rectified MeanFlow, a framework that models the mean velocity field along the rectified trajectory using only a single reflow step. This eliminates the need for perfectly straightened trajectories while enabling efficient training. Furthermore, we introduce a simple yet effective truncation heuristic that aims to reduce residual curvature and further improve performance. Extensive experiments on ImageNet at 64, 256, and 512 resolutions show that Re-MeanFlow consistently outperforms prior one-step flow distillation and Rectified Flow methods in both sample quality and training efficiency. Code is available at https://github.com/Xinxi-Zhang/Re-MeanFlow.

Recent AI-based 3D content creation has largely evolved along two paths: feed-forward image-to-3D reconstruction approaches and 3D generative models trained with 2D or 3D supervision. In this work, we show that existing feed-forward reconstruction methods can serve as effective latent encoders for training 3D generative models, thereby bridging these two paradigms. By reusing powerful pre-trained reconstruction models, we avoid computationally expensive encoder network training and obtain rich 3D latent features for generative modeling for free. However, the latent spaces of reconstruction models are not well-suited for generative modeling due to their unstructured nature. To enable flow-based model training on these latent features, we develop post-processing pipelines, including protocols to standardize the features and spatial weighting to concentrate on important regions. We further incorporate a 2D image space perceptual rendering loss to handle the high-dimensional latent spaces. Finally, we propose a multi-stream transformer-based rectified flow architecture to achieve linear scaling and high-quality text-conditioned 3D generation. Our framework leverages the advancements of feed-forward reconstruction models to enhance the scalability of 3D generative modeling, achieving both high computational efficiency and state-of-the-art performance in text-to-3D generation.

Flow-based text-to-image (T2I) models excel at prompt-driven image generation, but falter on Image Restoration (IR), often "drifting away" from being faithful to the measurement. Prior work mitigate this drift with data-specific flows or task-specific adapters that are computationally heavy and not scalable across tasks. This raises the question "Can't we efficiently manipulate the existing generative capabilities of a flow model?" To this end, we introduce FlowSteer (FS), an operator-aware conditioning scheme that injects measurement priors along the sampling path,coupling a frozed flow's implicit guidance with explicit measurement constraints. Across super-resolution, deblurring, denoising, and colorization, FS improves measurement consistency and identity preservation in a strictly zero-shot setting-no retrained models, no adapters. We show how the nature of flow models and their sensitivities to noise inform the design of such a scheduler. FlowSteer, although simple, achieves a higher fidelity of reconstructed images, while leveraging the rich generative priors of flow models. All data and code will be publicly available https://tharindu-nirmal.github.io/FlowSteer/{in this link}.

Normalizing flows (NFs) have become a prominent method for deep generative models that allow for an analytic probability density estimation and efficient synthesis. However, a flow-based network is considered to be inefficient in parameter complexity because of reduced expressiveness of bijective mapping, which renders the models unfeasibly expensive in terms of parameters. We present an alternative parameterization scheme called NanoFlow, which uses a single neural density estimator to model multiple transformation stages. Hence, we propose an efficient parameter decomposition method and the concept of flow indication embedding, which are key missing components that enable density estimation from a single neural network. Experiments performed on audio and image models confirm that our method provides a new parameter-efficient solution for scalable NFs with significant sublinear parameter complexity.

Recently, Flow Matching models have pushed the boundaries of high-fidelity data generation across a wide range of domains. It typically employs a single large network to learn the entire generative trajectory from noise to data. Despite their effectiveness, this design struggles to capture distinct signal characteristics across timesteps simultaneously and incurs substantial inference costs due to the iterative evaluation of the entire model. To address these limitations, we propose Blockwise Flow Matching (BFM), a novel framework that partitions the generative trajectory into multiple temporal segments, each modeled by smaller but specialized velocity blocks. This blockwise design enables each block to specialize effectively in its designated interval, improving inference efficiency and sample quality. To further enhance generation fidelity, we introduce a Semantic Feature Guidance module that explicitly conditions velocity blocks on semantically rich features aligned with pretrained representations. Additionally, we propose a lightweight Feature Residual Approximation strategy that preserves semantic quality while significantly reducing inference cost. Extensive experiments on ImageNet 256x256 demonstrate that BFM establishes a substantially improved Pareto frontier over existing Flow Matching methods, achieving 2.1x to 4.9x accelerations in inference complexity at comparable generation performance. Code is available at https://github.com/mlvlab/BFM.

ComfyUI provides a widely-adopted, workflow-based interface that enables users to customize various image generation tasks through an intuitive node-based architecture. However, the intricate connections between nodes and diverse modules often present a steep learning curve for users. In this paper, we introduce ComfyGPT, the first self-optimizing multi-agent system designed to generate ComfyUI workflows based on task descriptions automatically. ComfyGPT comprises four specialized agents: ReformatAgent, FlowAgent, RefineAgent, and ExecuteAgent. The core innovation of ComfyGPT lies in two key aspects. First, it focuses on generating individual node links rather than entire workflows, significantly improving generation precision. Second, we proposed FlowAgent, a LLM-based workflow generation agent that uses both supervised fine-tuning (SFT) and reinforcement learning (RL) to improve workflow generation accuracy. Moreover, we introduce FlowDataset, a large-scale dataset containing 13,571 workflow-description pairs, and FlowBench, a comprehensive benchmark for evaluating workflow generation systems. We also propose four novel evaluation metrics: Format Validation (FV), Pass Accuracy (PA), Pass Instruct Alignment (PIA), and Pass Node Diversity (PND). Experimental results demonstrate that ComfyGPT significantly outperforms existing LLM-based methods in workflow generation.

Diffusion and flow models have become the dominant paradigm for generative modeling on Riemannian manifolds, with successful applications in protein backbone generation and DNA sequence design. However, these methods require tens to hundreds of neural network evaluations at inference time, which can become a computational bottleneck in large-scale scientific sampling workflows. We introduce Riemannian MeanFlow~(RMF), a framework for learning flow maps directly on manifolds, enabling high-quality generations with as few as one forward pass. We derive three equivalent characterizations of the manifold average velocity (Eulerian, Lagrangian, and semigroup identities), and analyze parameterizations and stabilization techniques to improve training on high-dimensional manifolds. In promoter DNA design and protein backbone generation settings, RMF achieves comparable sample quality to prior methods while requiring up to 10times fewer function evaluations. Finally, we show that few-step flow maps enable efficient reward-guided design through reward look-ahead, where terminal states can be predicted from intermediate steps at minimal additional cost.

Modern generative models possess a deep understanding of visual content, yet training them for image editing typically requires massive datasets of paired examples. This limits scalability, especially for video editing where collecting paired data is prohibitively expensive. We propose Bootstrap Your Generator (ByG), a general framework for unpaired training of flow matching editing models. It leverages the base model's knowledge without any external signal. Our approach pairs instruction-following cues extracted from the frozen model with cycle-consistency for structure preservation. To make this tractable, we propose to route gradients from downstream losses over clean predictions to noisy training states. We demonstrate state-of-the-art results on challenging data-scarce image and video editing scenarios. Extensive evaluations and user studies show that our method effectively generalizes to unseen domains and outperforms supervised baselines trained on millions of samples. Analysis reveals that our gradient routing bridges the train-inference gap, and extracting semantic cues from a base model provides a robust training signal that obviates the need for external reward models.

We present FrameFlow, a method for fast protein backbone generation using SE(3) flow matching. Specifically, we adapt FrameDiff, a state-of-the-art diffusion model, to the flow-matching generative modeling paradigm. We show how flow matching can be applied on SE(3) and propose modifications during training to effectively learn the vector field. Compared to FrameDiff, FrameFlow requires five times fewer sampling timesteps while achieving two fold better designability. The ability to generate high quality protein samples at a fraction of the cost of previous methods paves the way towards more efficient generative models in de novo protein design.

This paper introduces Bayesian Flow Networks (BFNs), a new class of generative model in which the parameters of a set of independent distributions are modified with Bayesian inference in the light of noisy data samples, then passed as input to a neural network that outputs a second, interdependent distribution. Starting from a simple prior and iteratively updating the two distributions yields a generative procedure similar to the reverse process of diffusion models; however it is conceptually simpler in that no forward process is required. Discrete and continuous-time loss functions are derived for continuous, discretised and discrete data, along with sample generation procedures. Notably, the network inputs for discrete data lie on the probability simplex, and are therefore natively differentiable, paving the way for gradient-based sample guidance and few-step generation in discrete domains such as language modelling. The loss function directly optimises data compression and places no restrictions on the network architecture. In our experiments BFNs achieve competitive log-likelihoods for image modelling on dynamically binarized MNIST and CIFAR-10, and outperform all known discrete diffusion models on the text8 character-level language modelling task.

We introduce CaLMFlow (Causal Language Models for Flow Matching), a novel framework that casts flow matching as a Volterra integral equation (VIE), leveraging the power of large language models (LLMs) for continuous data generation. CaLMFlow enables the direct application of LLMs to learn complex flows by formulating flow matching as a sequence modeling task, bridging discrete language modeling and continuous generative modeling. Our method implements tokenization across space and time, thereby solving a VIE over these domains. This approach enables efficient handling of high-dimensional data and outperforms ODE solver-dependent methods like conditional flow matching (CFM). We demonstrate CaLMFlow's effectiveness on synthetic and real-world data, including single-cell perturbation response prediction, showcasing its ability to incorporate textual context and generalize to unseen conditions. Our results highlight LLM-driven flow matching as a promising paradigm in generative modeling, offering improved scalability, flexibility, and context-awareness.

We tackle the problem of sampling from intractable high-dimensional density functions, a fundamental task that often appears in machine learning and statistics. We extend recent sampling-based approaches that leverage controlled stochastic processes to model approximate samples from these target densities. The main drawback of these approaches is that the training objective requires full trajectories to compute, resulting in sluggish credit assignment issues due to use of entire trajectories and a learning signal present only at the terminal time. In this work, we present Diffusion Generative Flow Samplers (DGFS), a sampling-based framework where the learning process can be tractably broken down into short partial trajectory segments, via parameterizing an additional "flow function". Our method takes inspiration from the theory developed for generative flow networks (GFlowNets), allowing us to make use of intermediate learning signals. Through various challenging experiments, we demonstrate that DGFS achieves more accurate estimates of the normalization constant than closely-related prior methods.

Generative Flow Networks (GFlowNets) are amortized samplers that learn stochastic policies to sequentially generate compositional objects from a given unnormalized reward distribution. They can generate diverse sets of high-reward objects, which is an important consideration in scientific discovery tasks. However, as they are typically trained from a given extrinsic reward function, it remains an important open challenge about how to leverage the power of pre-training and train GFlowNets in an unsupervised fashion for efficient adaptation to downstream tasks. Inspired by recent successes of unsupervised pre-training in various domains, we introduce a novel approach for reward-free pre-training of GFlowNets. By framing the training as a self-supervised problem, we propose an outcome-conditioned GFlowNet (OC-GFN) that learns to explore the candidate space. Specifically, OC-GFN learns to reach any targeted outcomes, akin to goal-conditioned policies in reinforcement learning. We show that the pre-trained OC-GFN model can allow for a direct extraction of a policy capable of sampling from any new reward functions in downstream tasks. Nonetheless, adapting OC-GFN on a downstream task-specific reward involves an intractable marginalization over possible outcomes. We propose a novel way to approximate this marginalization by learning an amortized predictor enabling efficient fine-tuning. Extensive experimental results validate the efficacy of our approach, demonstrating the effectiveness of pre-training the OC-GFN, and its ability to swiftly adapt to downstream tasks and discover modes more efficiently. This work may serve as a foundation for further exploration of pre-training strategies in the context of GFlowNets.

Protein backbone generation plays a central role in de novo protein design and is significant for many biological and medical applications. Although diffusion and flow-based generative models provide potential solutions to this challenging task, they often generate proteins with undesired designability and suffer computational inefficiency. In this study, we propose a novel rectified quaternion flow (ReQFlow) matching method for fast and high-quality protein backbone generation. In particular, our method generates a local translation and a 3D rotation from random noise for each residue in a protein chain, which represents each 3D rotation as a unit quaternion and constructs its flow by spherical linear interpolation (SLERP) in an exponential format. We train the model by quaternion flow (QFlow) matching with guaranteed numerical stability and rectify the QFlow model to accelerate its inference and improve the designability of generated protein backbones, leading to the proposed ReQFlow model. Experiments show that ReQFlow achieves state-of-the-art performance in protein backbone generation while requiring much fewer sampling steps and significantly less inference time (e.g., being 37x faster than RFDiffusion and 62x faster than Genie2 when generating a backbone of length 300), demonstrating its effectiveness and efficiency. The code is available at https://github.com/AngxiaoYue/ReQFlow.

In this paper, we present gfnx, a fast and scalable package for training and evaluating Generative Flow Networks (GFlowNets) written in JAX. gfnx provides an extensive set of environments and metrics for benchmarking, accompanied with single-file implementations of core objectives for training GFlowNets. We include synthetic hypergrids, multiple sequence generation environments with various editing regimes and particular reward designs for molecular generation, phylogenetic tree construction, Bayesian structure learning, and sampling from the Ising model energy. Across different tasks, gfnx achieves significant wall-clock speedups compared to Pytorch-based benchmarks (such as torchgfn library) and author implementations. For example, gfnx achieves up to 55 times speedup on CPU-based sequence generation environments, and up to 80 times speedup with the GPU-based Bayesian network structure learning setup. Our package provides a diverse set of benchmarks and aims to standardize empirical evaluation and accelerate research and applications of GFlowNets. The library is available on GitHub (https://github.com/d-tiapkin/gfnx) and on pypi (https://pypi.org/project/gfnx/). Documentation is available on https://gfnx.readthedocs.io.

Flow-based generation in high-dimensional spaces is difficult because velocity prediction requires modeling high-dimensional noise, even when data has strong low-rank structure. We present Asymmetric Flow Modeling (AsymFlow), a rank-asymmetric velocity parameterization that restricts noise prediction to a low-rank subspace while keeping data prediction full-dimensional. From this asymmetric prediction, AsymFlow analytically recovers the full-dimensional velocity without changing the network architecture or training/sampling procedures. On ImageNet 256times256, AsymFlow achieves a leading 1.57 FID, outperforming prior DiT/JiT-like pixel diffusion models by a large margin. AsymFlow also provides the first-ever route for finetuning pretrained latent flow models into pixel-space models: aligning the low-rank pixel subspace to the latent space gives a seamless initialization that preserves the latent model's high-level semantics and structure, so finetuning mainly improves low-level mismatches rather than relearning pixel generation. We show that the pixel AsymFlow model finetuned from FLUX.2 klein 9B establishes a new state of the art for pixel-space text-to-image generation, beating its latent base on HPSv3, DPG-Bench, and GenEval while qualitatively showing substantially improved visual realism.

Flow Matching (FM) is a recent framework for generative modeling that has achieved state-of-the-art performance across various domains, including image, video, audio, speech, and biological structures. This guide offers a comprehensive and self-contained review of FM, covering its mathematical foundations, design choices, and extensions. By also providing a PyTorch package featuring relevant examples (e.g., image and text generation), this work aims to serve as a resource for both novice and experienced researchers interested in understanding, applying and further developing FM.

The growing popularity of generative flow networks (GFlowNets or GFNs) from a range of researchers with diverse backgrounds and areas of expertise necessitates a library that facilitates the testing of new features (e.g., training losses and training policies) against standard benchmark implementations, or on a set of common environments. We present torchgfn, a PyTorch library that aims to address this need. Its core contribution is a modular and decoupled architecture which treats environments, neural network modules, and training objectives as interchangeable components. This provides users with a simple yet powerful API to facilitate rapid prototyping and novel research. Multiple examples are provided, replicating and unifying published results. The library is available on GitHub (https://github.com/GFNOrg/torchgfn) and on pypi (https://pypi.org/project/torchgfn/).

Transformers based on the attention mechanism have achieved impressive success in various areas. However, the attention mechanism has a quadratic complexity, significantly impeding Transformers from dealing with numerous tokens and scaling up to bigger models. Previous methods mainly utilize the similarity decomposition and the associativity of matrix multiplication to devise linear-time attention mechanisms. They avoid degeneration of attention to a trivial distribution by reintroducing inductive biases such as the locality, thereby at the expense of model generality and expressiveness. In this paper, we linearize Transformers free from specific inductive biases based on the flow network theory. We cast attention as the information flow aggregated from the sources (values) to the sinks (results) through the learned flow capacities (attentions). Within this framework, we apply the property of flow conservation into attention and propose the Flow-Attention mechanism of linear complexity. By respectively conserving the incoming flow of sinks for source competition and the outgoing flow of sources for sink allocation, Flow-Attention inherently generates informative attentions without using specific inductive biases. Empowered by the Flow-Attention, Flowformer yields strong performance in linear time for wide areas, including long sequence, time series, vision, natural language, and reinforcement learning. The code and settings are available at this repository: https://github.com/thuml/Flowformer.

Smart meter data is the foundation for planning and operating the distribution network. Unfortunately, such data are not always available due to privacy regulations. Meanwhile, the collected data may be corrupted due to sensor or transmission failure, or it may not have sufficient resolution for downstream tasks. A wide range of generative tasks is formulated to address these issues, including synthetic data generation, missing data imputation, and super-resolution. Despite the success of machine learning models on these tasks, dedicated models need to be designed and trained for each task, leading to redundancy and inefficiency. In this paper, by recognizing the powerful modeling capability of flow matching models, we propose a new approach to unify diverse smart meter data generative tasks with a single model trained for conditional generation. The proposed flow matching models are trained to generate challenging, high-dimensional time series data, specifically monthly smart meter data at a 15 min resolution. By viewing different generative tasks as distinct forms of partial data observations and injecting them into the generation process, we unify tasks such as imputation and super-resolution with a single model, eliminating the need for re-training. The data generated by our model not only are consistent with the given observations but also remain realistic, showing better performance against interpolation and other machine learning based baselines dedicated to the tasks.

This monograph presents the core principles that have guided the development of diffusion models, tracing their origins and showing how diverse formulations arise from shared mathematical ideas. Diffusion modeling starts by defining a forward process that gradually corrupts data into noise, linking the data distribution to a simple prior through a continuum of intermediate distributions. The goal is to learn a reverse process that transforms noise back into data while recovering the same intermediates. We describe three complementary views. The variational view, inspired by variational autoencoders, sees diffusion as learning to remove noise step by step. The score-based view, rooted in energy-based modeling, learns the gradient of the evolving data distribution, indicating how to nudge samples toward more likely regions. The flow-based view, related to normalizing flows, treats generation as following a smooth path that moves samples from noise to data under a learned velocity field. These perspectives share a common backbone: a time-dependent velocity field whose flow transports a simple prior to the data. Sampling then amounts to solving a differential equation that evolves noise into data along a continuous trajectory. On this foundation, the monograph discusses guidance for controllable generation, efficient numerical solvers, and diffusion-motivated flow-map models that learn direct mappings between arbitrary times. It provides a conceptual and mathematically grounded understanding of diffusion models for readers with basic deep-learning knowledge.

Flow-based Transformer models have achieved state-of-the-art image generation performance, but often suffer from high inference latency and computational cost due to their large parameter sizes. To improve inference efficiency without compromising quality, we propose Bridged Progressive Rectified Flow Transformers (NAMI), which decompose the generation process across temporal, spatial, and architectural demensions. We divide the rectified flow into different stages according to resolution, and use a BridgeFlow module to connect them. Fewer Transformer layers are used at low-resolution stages to generate image layouts and concept contours, and more layers are progressively added as the resolution increases. Experiments demonstrate that our approach achieves fast convergence and reduces inference time while ensuring generation quality. The main contributions of this paper are summarized as follows: (1) We introduce Bridged Progressive Rectified Flow Transformers that enable multi-resolution training, accelerating model convergence; (2) NAMI leverages piecewise flow and spatial cascading of Diffusion Transformer (DiT) to rapidly generate images, reducing inference time by 64% for generating 1024 resolution images; (3) We propose a BridgeFlow module to align flows between different stages; (4) We propose the NAMI-1K benchmark to evaluate human preference performance, aiming to mitigate distributional bias and comprehensively assess model effectiveness. The results show that our model is competitive with state-of-the-art models.

In this work, we introduce ChemBFN, a language model that handles chemistry tasks based on Bayesian flow networks working on discrete data. A new accuracy schedule is proposed to improve the sampling quality by significantly reducing the reconstruction loss. We show evidence that our method is appropriate for generating molecules with satisfied diversity even when a smaller number of sampling steps is used. A classifier-free guidance method is adapted for conditional generation. It is also worthwhile to point out that after generative training, our model can be fine-tuned on regression and classification tasks with the state-of-the-art performance, which opens the gate of building all-in-one models in a single module style. Our model has been open sourced at https://github.com/Augus1999/bayesian-flow-network-for-chemistry.

Generative flow networks (GFlowNets) are a family of algorithms for training a sequential sampler of discrete objects under an unnormalized target density and have been successfully used for various probabilistic modeling tasks. Existing training objectives for GFlowNets are either local to states or transitions, or propagate a reward signal over an entire sampling trajectory. We argue that these alternatives represent opposite ends of a gradient bias-variance tradeoff and propose a way to exploit this tradeoff to mitigate its harmful effects. Inspired by the TD(lambda) algorithm in reinforcement learning, we introduce subtrajectory balance or SubTB(lambda), a GFlowNet training objective that can learn from partial action subsequences of varying lengths. We show that SubTB(lambda) accelerates sampler convergence in previously studied and new environments and enables training GFlowNets in environments with longer action sequences and sparser reward landscapes than what was possible before. We also perform a comparative analysis of stochastic gradient dynamics, shedding light on the bias-variance tradeoff in GFlowNet training and the advantages of subtrajectory balance.

We present JanusFlow, a powerful framework that unifies image understanding and generation in a single model. JanusFlow introduces a minimalist architecture that integrates autoregressive language models with rectified flow, a state-of-the-art method in generative modeling. Our key finding demonstrates that rectified flow can be straightforwardly trained within the large language model framework, eliminating the need for complex architectural modifications. To further improve the performance of our unified model, we adopt two key strategies: (i) decoupling the understanding and generation encoders, and (ii) aligning their representations during unified training. Extensive experiments show that JanusFlow achieves comparable or superior performance to specialized models in their respective domains, while significantly outperforming existing unified approaches across standard benchmarks. This work represents a step toward more efficient and versatile vision-language models.

Conditional generative modeling aims to learn a conditional data distribution from samples containing data-condition pairs. For this, diffusion and flow-based methods have attained compelling results. These methods use a learned (flow) model to transport an initial standard Gaussian noise that ignores the condition to the conditional data distribution. The model is hence required to learn both mass transport and conditional injection. To ease the demand on the model, we propose Condition-Aware Reparameterization for Flow Matching (CAR-Flow) -- a lightweight, learned shift that conditions the source, the target, or both distributions. By relocating these distributions, CAR-Flow shortens the probability path the model must learn, leading to faster training in practice. On low-dimensional synthetic data, we visualize and quantify the effects of CAR. On higher-dimensional natural image data (ImageNet-256), equipping SiT-XL/2 with CAR-Flow reduces FID from 2.07 to 1.68, while introducing less than 0.6% additional parameters.

The rapidly evolving field of Robotic Process Automation (RPA) has made significant strides in automating repetitive processes, yet its effectiveness diminishes in scenarios requiring spontaneous or unpredictable tasks demanded by users. This paper introduces a novel approach, FlowMind, leveraging the capabilities of Large Language Models (LLMs) such as Generative Pretrained Transformer (GPT), to address this limitation and create an automatic workflow generation system. In FlowMind, we propose a generic prompt recipe for a lecture that helps ground LLM reasoning with reliable Application Programming Interfaces (APIs). With this, FlowMind not only mitigates the common issue of hallucinations in LLMs, but also eliminates direct interaction between LLMs and proprietary data or code, thus ensuring the integrity and confidentiality of information - a cornerstone in financial services. FlowMind further simplifies user interaction by presenting high-level descriptions of auto-generated workflows, enabling users to inspect and provide feedback effectively. We also introduce NCEN-QA, a new dataset in finance for benchmarking question-answering tasks from N-CEN reports on funds. We used NCEN-QA to evaluate the performance of workflows generated by FlowMind against baseline and ablation variants of FlowMind. We demonstrate the success of FlowMind, the importance of each component in the proposed lecture recipe, and the effectiveness of user interaction and feedback in FlowMind.

This paper is about the problem of learning a stochastic policy for generating an object (like a molecular graph) from a sequence of actions, such that the probability of generating an object is proportional to a given positive reward for that object. Whereas standard return maximization tends to converge to a single return-maximizing sequence, there are cases where we would like to sample a diverse set of high-return solutions. These arise, for example, in black-box function optimization when few rounds are possible, each with large batches of queries, where the batches should be diverse, e.g., in the design of new molecules. One can also see this as a problem of approximately converting an energy function to a generative distribution. While MCMC methods can achieve that, they are expensive and generally only perform local exploration. Instead, training a generative policy amortizes the cost of search during training and yields to fast generation. Using insights from Temporal Difference learning, we propose GFlowNet, based on a view of the generative process as a flow network, making it possible to handle the tricky case where different trajectories can yield the same final state, e.g., there are many ways to sequentially add atoms to generate some molecular graph. We cast the set of trajectories as a flow and convert the flow consistency equations into a learning objective, akin to the casting of the Bellman equations into Temporal Difference methods. We prove that any global minimum of the proposed objectives yields a policy which samples from the desired distribution, and demonstrate the improved performance and diversity of GFlowNet on a simple domain where there are many modes to the reward function, and on a molecule synthesis task.

The remarkable success of diffusion and flow-matching models has ignited a surge of works on adapting them at test time for controlled generation tasks. Examples range from image editing to restoration, compression and personalization. However, due to the iterative nature of the sampling process in those models, it is computationally impractical to use gradient-based optimization to directly control the image generated at the end of the process. As a result, existing methods typically resort to manipulating each timestep separately. Here we introduce FlowOpt - a zero-order (gradient-free) optimization framework that treats the entire flow process as a black box, enabling optimization through the whole sampling path without backpropagation through the model. Our method is both highly efficient and allows users to monitor the intermediate optimization results and perform early stopping if desired. We prove a sufficient condition on FlowOpt's step-size, under which convergence to the global optimum is guaranteed. We further show how to empirically estimate this upper bound so as to choose an appropriate step-size. We demonstrate how FlowOpt can be used for image editing, showcasing two options: (i) inversion (determining the initial noise that generates a given image), and (ii) directly steering the edited image to be similar to the source image while conforming to a target text prompt. In both cases, FlowOpt achieves state-of-the-art results while using roughly the same number of neural function evaluations (NFEs) as existing methods. Code and examples are available on the project's webpage.

Autoregressive (AR) modeling has achieved remarkable success in natural language processing by enabling models to generate text with coherence and contextual understanding through next token prediction. Recently, in image generation, VAR proposes scale-wise autoregressive modeling, which extends the next token prediction to the next scale prediction, preserving the 2D structure of images. However, VAR encounters two primary challenges: (1) its complex and rigid scale design limits generalization in next scale prediction, and (2) the generator's dependence on a discrete tokenizer with the same complex scale structure restricts modularity and flexibility in updating the tokenizer. To address these limitations, we introduce FlowAR, a general next scale prediction method featuring a streamlined scale design, where each subsequent scale is simply double the previous one. This eliminates the need for VAR's intricate multi-scale residual tokenizer and enables the use of any off-the-shelf Variational AutoEncoder (VAE). Our simplified design enhances generalization in next scale prediction and facilitates the integration of Flow Matching for high-quality image synthesis. We validate the effectiveness of FlowAR on the challenging ImageNet-256 benchmark, demonstrating superior generation performance compared to previous methods. Codes will be available at https://github.com/OliverRensu/FlowAR.

Few-step generation has been a long-standing goal, with recent one-step generation methods exemplified by MeanFlow achieving remarkable results. Existing research on MeanFlow primarily focuses on class-to-image generation. However, an intuitive yet unexplored direction is to extend the condition from fixed class labels to flexible text inputs, enabling richer content creation. Compared to the limited class labels, text conditions pose greater challenges to the model's understanding capability, necessitating the effective integration of powerful text encoders into the MeanFlow framework. Surprisingly, although incorporating text conditions appears straightforward, we find that integrating powerful LLM-based text encoders using conventional training strategies results in unsatisfactory performance. To uncover the underlying cause, we conduct detailed analyses and reveal that, due to the extremely limited number of refinement steps in the MeanFlow generation, such as only one step, the text feature representations are required to possess sufficiently high discriminability. This also explains why discrete and easily distinguishable class features perform well within the MeanFlow framework. Guided by these insights, we leverage a powerful LLM-based text encoder validated to possess the required semantic properties and adapt the MeanFlow generation process to this framework, resulting in efficient text-conditioned synthesis for the first time. Furthermore, we validate our approach on the widely used diffusion model, demonstrating significant generation performance improvements. We hope this work provides a general and practical reference for future research on text-conditioned MeanFlow generation. The code is available at https://github.com/AMAP-ML/EMF.

Generative flow networks (GFlowNets) are a method for learning a stochastic policy for generating compositional objects, such as graphs or strings, from a given unnormalized density by sequences of actions, where many possible action sequences may lead to the same object. We find previously proposed learning objectives for GFlowNets, flow matching and detailed balance, which are analogous to temporal difference learning, to be prone to inefficient credit propagation across long action sequences. We thus propose a new learning objective for GFlowNets, trajectory balance, as a more efficient alternative to previously used objectives. We prove that any global minimizer of the trajectory balance objective can define a policy that samples exactly from the target distribution. In experiments on four distinct domains, we empirically demonstrate the benefits of the trajectory balance objective for GFlowNet convergence, diversity of generated samples, and robustness to long action sequences and large action spaces.

Flow matching has recently emerged as a powerful alternative to diffusion models, providing a continuous-time formulation for generative modeling and representation learning. Yet, we show that this framework suffers from a fundamental instability in the low-noise regime. As noise levels approach zero, arbitrarily small perturbations in the input can induce large variations in the velocity target, causing the condition number of the learning problem to diverge. This ill-conditioning not only slows optimization but also forces the encoder to reallocate its limited Jacobian capacity toward noise directions, thereby degrading semantic representations. We provide the first theoretical analysis of this phenomenon, which we term the low-noise pathology, establishing its intrinsic link to the structure of the flow matching objective. Building on these insights, we propose Local Contrastive Flow (LCF), a hybrid training protocol that replaces direct velocity regression with contrastive feature alignment at small noise levels, while retaining standard flow matching at moderate and high noise. Empirically, LCF not only improves convergence speed but also stabilizes representation quality. Our findings highlight the critical importance of addressing low-noise pathologies to unlock the full potential of flow matching for both generation and representation learning.

We present OneFlow, the first non-autoregressive multimodal model that enables variable-length and concurrent mixed-modal generation. Unlike autoregressive models that enforce rigid causal ordering between text and image generation, OneFlow combines an insertion-based Edit Flow for discrete text tokens with Flow Matching for image latents. OneFlow enables concurrent text-image synthesis with hierarchical sampling that prioritizes content over grammar. Through controlled experiments across model sizes from 1B to 8B, we demonstrate that OneFlow outperforms autoregressive baselines on both generation and understanding tasks while using up to 50% fewer training FLOPs. OneFlow surpasses both autoregressive and diffusion-based approaches while unlocking new capabilities for concurrent generation, iterative refinement, and natural reasoning-like generation.

We introduce Lagrangian Flow Networks (LFlows) for modeling fluid densities and velocities continuously in space and time. By construction, the proposed LFlows satisfy the continuity equation, a PDE describing mass conservation in its differentiable form. Our model is based on the insight that solutions to the continuity equation can be expressed as time-dependent density transformations via differentiable and invertible maps. This follows from classical theory of the existence and uniqueness of Lagrangian flows for smooth vector fields. Hence, we model fluid densities by transforming a base density with parameterized diffeomorphisms conditioned on time. The key benefit compared to methods relying on numerical ODE solvers or PINNs is that the analytic expression of the velocity is always consistent with changes in density. Furthermore, we require neither expensive numerical solvers, nor additional penalties to enforce the PDE. LFlows show higher predictive accuracy in density modeling tasks compared to competing models in 2D and 3D, while being computationally efficient. As a real-world application, we model bird migration based on sparse weather radar measurements.

Flow-based generative models have shown remarkable success in text-to-image generation, yet fine-tuning them with intermediate feedback remains challenging, especially for continuous-time flow matching models. Most existing approaches solely learn from outcome rewards, struggling with the credit assignment problem. Alternative methods that attempt to learn a critic via direct regression on cumulative rewards often face training instabilities and model collapse in online settings. We present AC-Flow, a robust actor-critic framework that addresses these challenges through three key innovations: (1) reward shaping that provides well-normalized learning signals to enable stable intermediate value learning and gradient control, (2) a novel dual-stability mechanism that combines advantage clipping to prevent destructive policy updates with a warm-up phase that allows the critic to mature before influencing the actor, and (3) a scalable generalized critic weighting scheme that extends traditional reward-weighted methods while preserving model diversity through Wasserstein regularization. Through extensive experiments on Stable Diffusion 3, we demonstrate that AC-Flow achieves state-of-the-art performance in text-to-image alignment tasks and generalization to unseen human preference models. Our results demonstrate that even with a computationally efficient critic model, we can robustly finetune flow models without compromising generative quality, diversity, or stability.

In recent years, trace generation has emerged as a significant challenge within the Process Mining community. Deep Learning (DL) models have demonstrated accuracy in reproducing the features of the selected processes. However, current DL generative models are limited in their ability to adapt the learned distributions to generate data samples based on specific conditions or attributes. This limitation is particularly significant because the ability to control the type of generated data can be beneficial in various contexts, enabling a focus on specific behaviours, exploration of infrequent patterns, or simulation of alternative 'what-if' scenarios. In this work, we address this challenge by introducing a conditional model for process data generation based on a conditional variational autoencoder (CVAE). Conditional models offer control over the generation process by tuning input conditional variables, enabling more targeted and controlled data generation. Unlike other domains, CVAE for process mining faces specific challenges due to the multiperspective nature of the data and the need to adhere to control-flow rules while ensuring data variability. Specifically, we focus on generating process executions conditioned on control flow and temporal features of the trace, allowing us to produce traces for specific, identified sub-processes. The generated traces are then evaluated using common metrics for generative model assessment, along with additional metrics to evaluate the quality of the conditional generation

Meshes are among the most common 3D scene representations, but directly generating meshes is challenging because the representation contains important symmetries, including permutation invariance of faces and vertices. MeshFlow learns to generate triangle meshes directly as triangle soups, avoiding the need to serialize meshes into long autoregressive sequences. We adopt equivariant optimal-transport flow matching models that respect the key symmetries of triangle soups: arbitrary permutations of faces and permutations of the vertices within each face. Toward this goal, we propose a simple yet effective modification to the Diffusion Transformer architecture, resulting in a scalable network capable of modeling a velocity field while maintaining the desired equivariance. We further introduce an optimal-transport-based training objective that improves convergence by eliminating supervision signals that violate these symmetries. MeshFlow achieves mesh quality comparable to state-of-the-art autoregressive mesh generators while providing about an 18times speedup during inference. Project page is at https://qiisun.github.io/MeshFlow/.

The discovery of extremal structures in mathematics requires navigating vast and nonconvex landscapes where analytical methods offer little guidance and brute-force search becomes intractable. We introduce FlowBoost, a closed-loop generative framework that learns to discover rare and extremal geometric structures by combining three components: (i) a geometry-aware conditional flow-matching model that learns to sample high-quality configurations, (ii) reward-guided policy optimization with action exploration that directly optimizes the generation process toward the objective while maintaining diversity, and (iii) stochastic local search for both training-data generation and final refinement. Unlike prior open-loop approaches, such as PatternBoost that retrains on filtered discrete samples, or AlphaEvolve which relies on frozen Large Language Models (LLMs) as evolutionary mutation operators, FlowBoost enforces geometric feasibility during sampling, and propagates reward signal directly into the generative model, closing the optimization loop and requiring much smaller training sets and shorter training times, and reducing the required outer-loop iterations by orders of magnitude, while eliminating dependence on LLMs. We demonstrate the framework on four geometric optimization problems: sphere packing in hypercubes, circle packing maximizing sum of radii, the Heilbronn triangle problem, and star discrepancy minimization. In several cases, FlowBoost discovers configurations that match or exceed the best known results. For circle packings, we improve the best known lower bounds, surpassing the LLM-based system AlphaEvolve while using substantially fewer computational resources.

We study the problem of training a flow-based generative model, parametrized by a two-layer autoencoder, to sample from a high-dimensional Gaussian mixture. We provide a sharp end-to-end analysis of the problem. First, we provide a tight closed-form characterization of the learnt velocity field, when parametrized by a shallow denoising auto-encoder trained on a finite number n of samples from the target distribution. Building on this analysis, we provide a sharp description of the corresponding generative flow, which pushes the base Gaussian density forward to an approximation of the target density. In particular, we provide closed-form formulae for the distance between the mean of the generated mixture and the mean of the target mixture, which we show decays as Theta_n(1{n}). Finally, this rate is shown to be in fact Bayes-optimal.

Automatic workflow generation is the process of automatically synthesizing sequences of LLM calls, tool invocations, and post-processing steps for complex end-to-end tasks. Most prior methods cast this task as an optimization problem with limited theoretical grounding. We propose to cast workflow generation as Bayesian inference over a posterior distribution on workflows, and introduce Bayesian Workflow Generation (BWG), a sampling framework that builds workflows step-by-step using parallel look-ahead rollouts for importance weighting and a sequential in-loop refiner for pool-wide improvements. We prove that, without the refiner, the weighted empirical distribution converges to the target posterior. We instantiate BWG as BayesFlow, a training-free algorithm for workflow construction. Across six benchmark datasets, BayesFlow improves accuracy by up to 9 percentage points over SOTA workflow generation baselines and by up to 65 percentage points over zero-shot prompting, establishing BWG as a principled upgrade to search-based workflow design. Code will be available on https://github.com/BoYuanVisionary/BayesFlow.

Generative models have demonstrated remarkable success in domains such as text, image, and video synthesis. In this work, we explore the application of generative models to fluid dynamics, specifically for turbulence simulation, where classical numerical solvers are computationally expensive. We propose a novel stochastic generative model based on stochastic interpolants, which enables probabilistic forecasting while incorporating physical constraints such as energy stability and divergence-freeness. Unlike conventional stochastic generative models, which are often agnostic to underlying physical laws, our approach embeds energy consistency by making the parameters of the stochastic interpolant learnable coefficients. We evaluate our method on a benchmark turbulence problem - Kolmogorov flow - demonstrating superior accuracy and stability over state-of-the-art alternatives such as autoregressive conditional diffusion models (ACDMs) and PDE-Refiner. Furthermore, we achieve stable results for significantly longer roll-outs than standard stochastic interpolants. Our results highlight the potential of physics-aware generative models in accelerating and enhancing turbulence simulations while preserving fundamental conservation properties.

Existing generative models for 3D shapes can synthesize high-fidelity and visually plausible shapes. For certain classes of shapes that have undergone an engineering design process, the realism of the shape is tightly coupled with the underlying physical properties, e.g., aerodynamic efficiency for automobiles. Since existing methods lack knowledge of such physics, they are unable to use this knowledge to enhance the realism of shape generation. Motivated by this, we propose a unified physics-based 3D shape generation pipeline, with a focus on industrial design applications. Specifically, we introduce a new flow matching model with explicit physical guidance, consisting of an alternating update process. We iteratively perform a velocity-based update and a physics-based refinement, progressively adjusting the latent code to align with the desired 3D shapes and physical properties. We further strengthen physical validity by incorporating a physics-aware regularization term into the velocity-based update step. To support such physics-guided updates, we build a shape-and-physics variational autoencoder (SP-VAE) that jointly encodes shape and physics information into a unified latent space. The experiments on three benchmarks show that this synergistic formulation improves shape realism beyond mere visual plausibility.

Most modern text-to-speech architectures use a WaveNet vocoder for synthesizing high-fidelity waveform audio, but there have been limitations, such as high inference time, in its practical application due to its ancestral sampling scheme. The recently suggested Parallel WaveNet and ClariNet have achieved real-time audio synthesis capability by incorporating inverse autoregressive flow for parallel sampling. However, these approaches require a two-stage training pipeline with a well-trained teacher network and can only produce natural sound by using probability distillation along with auxiliary loss terms. We propose FloWaveNet, a flow-based generative model for raw audio synthesis. FloWaveNet requires only a single-stage training procedure and a single maximum likelihood loss, without any additional auxiliary terms, and it is inherently parallel due to the characteristics of generative flow. The model can efficiently sample raw audio in real-time, with clarity comparable to previous two-stage parallel models. The code and samples for all models, including our FloWaveNet, are publicly available.