Generative diffusion and flow models are constructed to remain exactly on the Lorentz-invariant massless N-particle phase space manifold during sampling for particle physics applications.
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Flow Matching Guide and Code
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abstract
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.
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representative citing papers
A control-theoretic linear program yields value-driven transport policies for generative modeling with straight paths and simulation-free training.
DiSI disentangles stochastic interpolants into separate generation and regression paths, allowing controllable transitions between regression and generative image restoration with a unified few-step sampler.
Discrete MeanFlow parameterizes CTMC conditional transition kernels with a boundary-by-construction design to enable exact one-step generation in discrete state spaces.
FlowIQN is a quantile-coupled CFM critic that yields the first explicit Wasserstein-aligned approximate projection for distributional RL, with improved return-distribution accuracy and competitive offline RL performance.
MPFM models flow matching velocity as a Gaussian mixture prior per normal class plus a mutual information regularizer to improve open-set anomaly detection over unimodal prototypes.
OGPP is a particle flow-matching method using orbit-space canonicalization and geometric paths that achieves lower error and fewer steps than prior approaches on 3D benchmarks.
Binomial flows close the gap between continuous flow matching and discrete ordinal data by using binomial distributions to enable unified denoising, sampling, and exact likelihoods in diffusion models.
LeapAlign fine-tunes flow matching models by constructing two consecutive leaps that skip multiple ODE steps with randomized timesteps and consistency weighting, enabling stable updates at any generation step.
TokenLight encodes lighting attributes as tokens in a conditional image generation model trained mostly on synthetic data, enabling precise relighting control and implicit learning of light-scene interactions.
DoMinO reformulates discrete flow matching sampling as an MDP for unbiased RL fine-tuning with new TV regularizers, yielding better enhancer activity and naturalness on DNA design tasks.
DiNa-LRM introduces a diffusion-native latent reward model using a noise-calibrated Thurstone likelihood on noisy states, matching VLM performance at lower compute in image alignment and preference optimization.
Flow matching on time series targets a closed-form nonparametric velocity field that is a similarity-weighted mixture of observed transition velocities, making neural models approximations to an ideal memory-augmented dynamical system sampler.
Empirical flow matching introduces coupled biases from plug-in estimation, including altered statistical targets, non-gradient minimizers, and non-unique dynamics via flux-null fields, with base distribution controlling kinetic energy tails.
Béz ierFlow parameterizes stochastic interpolant schedulers as Béz ier functions to learn optimal sampling trajectories, achieving 2-3x better few-step performance than prior timestep optimization methods.
Reformulates constrained black-box optimization as posterior inference in latent space of flow-based models amortized by outsourced diffusion models, claiming superior performance on synthetic and real tasks.
Part²GS introduces a part-aware 3D Gaussian representation with physics-guided motion constraints and a repel point field for high-fidelity modeling of articulated objects.
DSRL steers pretrained diffusion policies for robotics by applying RL to their latent noise inputs, achieving sample-efficient real-world adaptation with only black-box access.
Analytic solution of full-batch gradient flow for linear and convolutional denoisers in diffusion models yields a universal inverse-variance spectral law for learning times of eigenmodes.
Sundial uses TimeFlow Loss for native pre-training of Transformers on continuous time series from TimeBench, achieving SOTA point and probabilistic forecasting with millisecond inference.
SITN performs single-sample OOD detection via goodness-of-fit testing on noise samples in the factorised latent space of continuous normalizing flows.
Lance presents a dual-stream mixture-of-experts model with modality-aware positional encoding and staged multi-task training that outperforms prior open-source unified models on image and video generation while keeping strong understanding performance.
Wavelet Flow Matching emulates multi-scale PDE-governed systems by transporting velocities directly in a hierarchical wavelet representation via U-Net, yielding improved long-horizon stability and spectral accuracy on fluid benchmarks.
TOPOS creates high-fidelity 3D heads with fixed industry topology from single images via a specialized VAE with Perceiver Resampler and a rectified flow transformer.
citing papers explorer
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Generative models on phase space
Generative diffusion and flow models are constructed to remain exactly on the Lorentz-invariant massless N-particle phase space manifold during sampling for particle physics applications.
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Generative Modeling by Value-Driven Transport
A control-theoretic linear program yields value-driven transport policies for generative modeling with straight paths and simulation-free training.
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Disentangling Generation and Regression in Stochastic Interpolants for Controllable Image Restoration
DiSI disentangles stochastic interpolants into separate generation and regression paths, allowing controllable transitions between regression and generative image restoration with a unified few-step sampler.
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Discrete MeanFlow: One-Step Generation via Conditional Transition Kernels
Discrete MeanFlow parameterizes CTMC conditional transition kernels with a boundary-by-construction design to enable exact one-step generation in discrete state spaces.
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Quantile-Coupled Flow Matching for Distributional Reinforcement Learning
FlowIQN is a quantile-coupled CFM critic that yields the first explicit Wasserstein-aligned approximate projection for distributional RL, with improved return-distribution accuracy and competitive offline RL performance.
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Mixture Prototype Flow Matching for Open-Set Supervised Anomaly Detection
MPFM models flow matching velocity as a Gaussian mixture prior per normal class plus a mutual information regularizer to improve open-set anomaly detection over unimodal prototypes.
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Generative Modeling with Orbit-Space Particle Flow Matching
OGPP is a particle flow-matching method using orbit-space canonicalization and geometric paths that achieves lower error and fewer steps than prior approaches on 3D benchmarks.
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Binomial flows: Denoising and flow matching for discrete ordinal data
Binomial flows close the gap between continuous flow matching and discrete ordinal data by using binomial distributions to enable unified denoising, sampling, and exact likelihoods in diffusion models.
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LeapAlign: Post-Training Flow Matching Models at Any Generation Step by Building Two-Step Trajectories
LeapAlign fine-tunes flow matching models by constructing two consecutive leaps that skip multiple ODE steps with randomized timesteps and consistency weighting, enabling stable updates at any generation step.
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TokenLight: Precise Lighting Control in Images using Attribute Tokens
TokenLight encodes lighting attributes as tokens in a conditional image generation model trained mostly on synthetic data, enabling precise relighting control and implicit learning of light-scene interactions.
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Discrete Flow Matching Policy Optimization
DoMinO reformulates discrete flow matching sampling as an MDP for unbiased RL fine-tuning with new TV regularizers, yielding better enhancer activity and naturalness on DNA design tasks.
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Beyond VLM-Based Rewards: Diffusion-Native Latent Reward Modeling
DiNa-LRM introduces a diffusion-native latent reward model using a noise-calibrated Thurstone likelihood on noisy states, matching VLM performance at lower compute in image alignment and preference optimization.
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Is Flow Matching Just Trajectory Replay for Sequential Data?
Flow matching on time series targets a closed-form nonparametric velocity field that is a similarity-weighted mixture of observed transition velocities, making neural models approximations to an ideal memory-augmented dynamical system sampler.
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On The Hidden Biases of Flow Matching Samplers
Empirical flow matching introduces coupled biases from plug-in estimation, including altered statistical targets, non-gradient minimizers, and non-unique dynamics via flux-null fields, with base distribution controlling kinetic energy tails.
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B\'ezierFlow: Learning B\'ezier Stochastic Interpolant Schedulers for Few-Step Generation
Béz ierFlow parameterizes stochastic interpolant schedulers as Béz ier functions to learn optimal sampling trajectories, achieving 2-3x better few-step performance than prior timestep optimization methods.
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Posterior Inference in Latent Space for Scalable Constrained Black-box Optimization
Reformulates constrained black-box optimization as posterior inference in latent space of flow-based models amortized by outsourced diffusion models, claiming superior performance on synthetic and real tasks.
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Part$^{2}$GS: Part-aware Modeling of Articulated Objects using 3D Gaussian Splatting
Part²GS introduces a part-aware 3D Gaussian representation with physics-guided motion constraints and a repel point field for high-fidelity modeling of articulated objects.
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Steering Your Diffusion Policy with Latent Space Reinforcement Learning
DSRL steers pretrained diffusion policies for robotics by applying RL to their latent noise inputs, achieving sample-efficient real-world adaptation with only black-box access.
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An Analytical Theory of Spectral Bias in the Learning Dynamics of Diffusion Models
Analytic solution of full-batch gradient flow for linear and convolutional denoisers in diffusion models yields a universal inverse-variance spectral law for learning times of eigenmodes.
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Sundial: A Family of Highly Capable Time Series Foundation Models
Sundial uses TimeFlow Loss for native pre-training of Transformers on continuous time series from TimeBench, achieving SOTA point and probabilistic forecasting with millisecond inference.
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The Signal in the Noise: OOD Detection Through Goodness-of-Fit Testing in Factorised Latent Spaces
SITN performs single-sample OOD detection via goodness-of-fit testing on noise samples in the factorised latent space of continuous normalizing flows.
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Lance: Unified Multimodal Modeling by Multi-Task Synergy
Lance presents a dual-stream mixture-of-experts model with modality-aware positional encoding and staged multi-task training that outperforms prior open-source unified models on image and video generation while keeping strong understanding performance.
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Wavelet Flow Matching for Multi-Scale Physics Emulation
Wavelet Flow Matching emulates multi-scale PDE-governed systems by transporting velocities directly in a hierarchical wavelet representation via U-Net, yielding improved long-horizon stability and spectral accuracy on fluid benchmarks.
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TOPOS: High-Fidelity and Efficient Industry-Grade 3D Head Generation
TOPOS creates high-fidelity 3D heads with fixed industry topology from single images via a specialized VAE with Perceiver Resampler and a rectified flow transformer.
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Discrete Flow Matching for Offline-to-Online Reinforcement Learning
DRIFT enables stable offline-to-online fine-tuning of CTMC policies in discrete RL via advantage-weighted discrete flow matching, path-space regularization, and candidate-set approximation.
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SF-Flow: Sound field magnitude estimation via flow matching guided by sparse measurements
SF-Flow applies flow matching with a permutation-invariant set encoder and 3D U-Net to reconstruct ATF magnitudes from sparse inputs, showing accurate results up to 1 kHz with faster training than autoencoder baselines.
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dFlowGRPO: Rate-Aware Policy Optimization for Discrete Flow Models
dFlowGRPO is a new rate-aware RL method for discrete flow models that outperforms prior GRPO approaches on image generation and matches continuous flow models while supporting broad probability paths.
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BRICKS: Compositional Neural Markov Kernels for Zero-Shot Radiation-Matter Simulation
BRICKS creates compositional neural Markov kernels via hybrid transformers and Riemannian Flow Matching on product manifolds to enable zero-shot simulation of radiation-matter interactions across arbitrary material distributions.
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A Few-Step Generative Model on Cumulative Flow Maps
Cumulative flow maps unify few-step generative modeling for diffusion and flow models via cumulative transport and parameterization with minimal changes to time embeddings and objectives.
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PRTS: A Primitive Reasoning and Tasking System via Contrastive Representations
PRTS pretrains VLA models with contrastive goal-conditioned RL to embed goal-reachability probabilities from offline data, yielding SOTA results on robotic benchmarks especially for long-horizon and novel instructions.
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Learning biophysical models of gene regulation with probability flow matching
Probability Flow Matching learns biophysically consistent stochastic processes for gene regulation from time-resolved single-cell measurements, where only the biophysical versions accurately capture lineage transitions, fate specification, and perturbation responses despite similar data fit.
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Occupancy Reward Shaping: Improving Credit Assignment for Offline Goal-Conditioned Reinforcement Learning
Occupancy Reward Shaping extracts goal-reaching rewards from world-model occupancy measures using optimal transport, improving offline goal-conditioned RL performance 2.2x on 13 tasks without changing the optimal policy.
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Fisher Decorator: Refining Flow Policy via a Local Transport Map
Fisher Decorator refines flow policies in offline RL via a local transport map and Fisher-matrix quadratic approximation of the KL constraint, yielding controllable error near the optimum and SOTA benchmark results.
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Towards Faster Language Model Inference Using Mixture-of-Experts Flow Matching
Mixture-of-experts flow matching enables non-autoregressive language models to achieve autoregressive-level quality in three sampling steps, delivering up to 1000x faster inference than diffusion models.
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PhyMix: Towards Physically Consistent Single-Image 3D Indoor Scene Generation with Implicit--Explicit Optimization
PhyMix unifies a new multi-aspect physics evaluator with implicit policy optimization and explicit test-time correction to produce single-image 3D indoor scenes that are both visually faithful and physically plausible.
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FASTER: Rethinking Real-Time Flow VLAs
FASTER adds a Horizon-Aware Schedule to flow VLAs that compresses immediate-action denoising to one step while keeping long-horizon trajectory quality, lowering real-robot reaction latency.
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Energy-Guided Generative Modeling for Low-Energy Molecular Structure Discovery
EnFlow integrates flow-based conformer generation with energy landscape modeling to enable joint ensemble generation and ground-state identification using only 1-2 ODE steps.
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SPEAR-1: Scaling Beyond Robot Demonstrations via 3D Understanding
SPEAR-1 combines a 3D-enriched VLM with embodied control to match or exceed existing robotic foundation models using 20 times fewer robot demonstrations.
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Energy-Weighted Flow Matching: Unlocking Continuous Normalizing Flows for Efficient and Scalable Boltzmann Sampling
Energy-Weighted Flow Matching reformulates conditional flow matching with importance sampling to enable continuous normalizing flows to model Boltzmann distributions from energy evaluations alone, with iterative and annealed variants showing competitive performance on benchmarks.
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HazeMatching: Dehazing Light Microscopy Images with Guided Conditional Flow Matching
HazeMatching adapts conditional flow matching with hazy-image guidance to dehaze microscopy images while balancing fidelity and realism on synthetic and real data.
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Mean Flows for One-step Generative Modeling
MeanFlow uses a derived identity between average and instantaneous velocities to train one-step flow models, achieving FID 3.43 on ImageNet 256x256 with 1-NFE from scratch.
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Abstraction for Offline Goal-Conditioned Reinforcement Learning
Introduces relativised options and hierarchical abstraction to reuse experience across similar contexts in offline GCRL, with two algorithms demonstrating performance gains.
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Drift Flow Matching
Drift Flow Matching connects direct transport maps from Drift Models with flow-based iterative refinement to enable adaptive computation in generative modeling.
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Sharpen Your Flow: Sharpness-Aware Sampling for Flow Matching
SharpEuler estimates a sharpness profile via finite differences on calibration trajectories, smooths it, and applies a quantile transform to generate adaptive timestep grids that improve Euler sampling quality in flow matching models at fixed budgets.
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A Stability Benchmark of Generative Regularizers for Inverse Problems
Numerical benchmarks indicate generative regularizers deliver strong reconstructions in some imaging inverse problem settings but can be unstable or problematic under imperfect conditions compared to variational methods.
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Deterministic Decomposition of Stochastic Generative Dynamics
Stochastic generative dynamics are decomposed into transport and osmotic parts via b_t = u_t + d_t, with Bridge Matching proposed to learn the components for controllable sampling.
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Exploring Time Conditioning in Diffusion Generative Models from Disjoint Noisy Data Manifolds
Aligning the DDIM forward diffusion process with flow-matching manifold evolution enables high-quality generation without time conditioning, and class-conditional synthesis is possible with an unconditional denoiser by using separate time spaces per class.
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Efficient Hierarchical Implicit Flow Q-learning for Offline Goal-conditioned Reinforcement Learning
Proposes mean flow policies and LeJEPA loss to overcome Gaussian policy limits and weak subgoal generation in hierarchical offline GCRL, reporting strong results on OGBench state and pixel tasks.
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Exploring Motion-Language Alignment for Text-driven Motion Generation
MLA-Gen advances text-driven motion synthesis by aligning global motion patterns with fine-grained text semantics and mitigating attention sink effects via new masking techniques.
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Woosh: A Sound Effects Foundation Model
Woosh is a new publicly released foundation model optimized for high-quality sound effect generation from text or video, showing competitive or better results than open alternatives like Stable Audio Open.