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
Proposes diffeomorphic optimization for manifold-constrained problems in generative models via flow maps, with Lie-group extensions for protein design showing metric improvements.
Flow-Map GRPO uses anchored stochastic flow map composition to enable GRPO-based RL alignment of deterministic few-step flow-map generators while preserving their marginal paths.
Introduces structured DRO for learned inverse problem reconstructions with ambiguity sets aligned to the forward operator, yielding explicit dual representations and a worst-case bound that induces Tikhonov regularization on the operator Lipschitz constant.
MATCH is the first flow matching method for multi-view anomaly detection, reporting SOTA results on Real-IAD and the first comprehensive evaluation on MANTA-Tiny while enabling real-time use by omitting the divergence term.
PAINT reframes asynchronous flow-based action chunking as an initial noise selection problem solved via backward Euler inversion and a repainting rule.
Self-distillation from a caption-conditioned video diffusion model to an image-and-prompt-conditioned executor, enhanced by RL from VLM feedback, enables task solving in world models.
ActFlow expands the generable set of pre-trained flow models for out-of-distribution molecular and sequence design via active synthetic data generation and verifier feedback, with new statistical guarantees.
RLDT fine-tunes pretrained flow-matching policies for continuous control by aligning them to a max-entropy RL transport field constructed via SVGD, using expected-target estimation for stable multi-step updates.
Low-Pass Flow Matching modifies Flow Matching via an operator-modulated interpolant inducing time-varying spectral bias from source spectrum to frequency-decaying bias, improving or preserving quality while reducing sampling cost on unconditional image generation including Galaxy10.
Introduces a state-aligned latent actor-critic framework that lets diffusion models act as their own timestep-conditioned value functions for trajectory-level RL post-training and inference steering.
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.
ALU uses public data to suppress unlearning cost quadratically while characterizing distribution mismatch effects, enabling mass unlearning with maintained utility.
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.
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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Diffeomorphic Optimization
Proposes diffeomorphic optimization for manifold-constrained problems in generative models via flow maps, with Lie-group extensions for protein design showing metric improvements.
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Flow-Map GRPO: Reinforcement Learning for Few-Step Flow-Map Generators via Anchored Stochastic Composition
Flow-Map GRPO uses anchored stochastic flow map composition to enable GRPO-based RL alignment of deterministic few-step flow-map generators while preserving their marginal paths.
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A Distributionally Robust Framework for Learned Reconstructions in Inverse Problems
Introduces structured DRO for learned inverse problem reconstructions with ambiguity sets aligned to the forward operator, yielding explicit dual representations and a worst-case bound that induces Tikhonov regularization on the operator Lipschitz constant.
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MATCH: Flow Matching for Multi-View Anomaly Detection
MATCH is the first flow matching method for multi-view anomaly detection, reporting SOTA results on Real-IAD and the first comprehensive evaluation on MANTA-Tiny while enabling real-time use by omitting the divergence term.
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Start Right, Arrive Right: Asynchronous Execution via Initial Noise Selection
PAINT reframes asynchronous flow-based action chunking as an initial noise selection problem solved via backward Euler inversion and a repainting rule.
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World Model Self-Distillation: Training World Models to Solve General Tasks
Self-distillation from a caption-conditioned video diffusion model to an image-and-prompt-conditioned executor, enhanced by RL from VLM feedback, enables task solving in world models.
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Active Flow Expansion for Out-of-Distribution Discovery: from Theory to Molecules
ActFlow expands the generable set of pre-trained flow models for out-of-distribution molecular and sequence design via active synthetic data generation and verifier feedback, with new statistical guarantees.
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Reinforcement Learning for Flow-Matching Policies with Density Transport
RLDT fine-tunes pretrained flow-matching policies for continuous control by aligning them to a max-entropy RL transport field constructed via SVGD, using expected-target estimation for stable multi-step updates.
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Low-Pass Flow Matching
Low-Pass Flow Matching modifies Flow Matching via an operator-modulated interpolant inducing time-varying spectral bias from source spectrum to frequency-decaying bias, improving or preserving quality while reducing sampling cost on unconditional image generation including Galaxy10.
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Explicit Critic Guidance for Aligning Diffusion Models
Introduces a state-aligned latent actor-critic framework that lets diffusion models act as their own timestep-conditioned value functions for trajectory-level RL post-training and inference steering.
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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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Unlearning with Asymmetric Sources: Improved Unlearning-Utility Trade-off with Public Data
ALU uses public data to suppress unlearning cost quadratically while characterizing distribution mismatch effects, enabling mass unlearning with maintained utility.
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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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Generative Model Proposal based Particle Filtering for Data Assimilation
FPPF learns a conditional flow-matching proposal that approximates the optimal particle-filter proposal, retains exact importance weights, and with localization outperforms classical and generative DA baselines on chaotic systems.
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Wavelet-Optimized Pseudo-3D Accelerated Diffusion Model for Truncated Computed Laminography
CL-DM combines 2D diffusion models with 3D MBIR, wavelet regularization, and fast sampling to eliminate truncation artifacts and restore continuous 3D structures in computed laminography beyond the FOV.
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Probabilistic Inversion with Flow Matching
Adapts Flow Matching from generative AI to probabilistic inversion, evaluated on a simple 2D velocity model and the OpenFWI seismic dataset.
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PerturbCellRL: Verifier-Guided Reinforcement Learning for Single-Cell Perturbation Prediction
PerturbCellRL is a reinforcement learning framework that post-trains single-cell transcriptomic generators using verifier rewards for improved biological consistency in perturbation predictions.
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The Geometry Behind Diffusion and Flow Matching: Gradient Flows and Geodesics in Wasserstein Space
Diffusion models realize JKO steps of free-energy gradient flow on Wasserstein space while flow matching follows Benamou-Brenier geodesics, unifying the two as initial-value vs boundary-value problems on the same manifold.
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A Time-Reparameterized Cumulative Intensity Extrapolation Sampler for Discrete Flow Matching
TR-CIE sampler improves DFM sampling quality under limited NFE via schedule-based time reparameterization and cumulative-intensity extrapolation without extra model calls.
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One-Step Flow Matching for Generative Modeling of Path-Dependent Physical Fields
A one-step flow matching model using transformer in VAE latent space with non-Gaussian source and auxiliary networks generates accurate high-resolution path-dependent stress fields, achieving 6-7x CPU and ~100x GPU speedup over FEM.
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Unsupervised Causal Abstractions Discovery
Low-rank graphs induce latents that form causal abstractions, with identifiability results and a practical objective enabling unsupervised learning of high-level SCMs from low-level measurements.
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Reversal Q-Learning
Reversal Q-Learning (RQL) proposes reversing flows for virtual trajectories and bias-variance reduction in an expanded MDP to train flow policies, reporting best average performance on 50 simulated robotic tasks versus prior flow-based offline RL methods.
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Perron--Frobenius Operator Matching for Generative Modeling
PFOM unifies generative models via Perron-Frobenius operator matching and proves KL is the only Bregman divergence equating density-level and sample-conditioned objectives.
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PianoKontext: Expressive Performance Rendering from Deadpan Context
PianoKontext renders expressive piano performances from deadpan scores using flow matching in Music2Latent latent space with DTW alignment for paired training data.
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Flexible Flows for Biological Sequence Design
Enhances Discrete Flow Matching with domain-specific couplings, latent edit-based rates, latent classifier-free guidance, and temperature scaling to reach SOTA on DNA and peptide sequence tasks.
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Latent Diffusion Policy: Shaping Latent Spaces for Diffusion-Based Robotic Manipulation
LDP shapes an observation-conditioned latent space with CVAE to simplify flow matching for diffusion policies, claiming substantial gains over DP3 on bimanual coordination tasks in simulation and real-world transfer.
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Spatiotemporal Imputation with Graph-Informed Flow Matching
GiFlow is a flow-matching model that builds a graph-informed prior via filtering of observable signals and uses a hybrid attention-propagation vector field to impute spatiotemporal data more accurately than prior methods.
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CellBRIDGE: Learning Cellular Trajectories via Interaction-Aware Alignment
CellBRIDGE augments feature-based optimal transport with a directed ligand-receptor interaction cost to improve cellular trajectory inference from scRNA-seq population snapshots.
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GenSBI: Generative Methods for Simulation-Based Inference in JAX
GenSBI delivers JAX-native implementations of generative SBI methods with transformer backbones and reports near-ideal calibration scores on standard benchmarks.
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Learning Energy-Based Models from Stochastic Interpolants using Spatiotemporal Differences
stNCE learns the energy of the joint density over data and time via spatiotemporal differences, unifies prior methods, and reports competitive performance on image and molecule density estimation.
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Guided Flow Matching for Forward and Inverse PDE Problems with Sparse Observations: Algorithm and Theory
FM4PDE applies flow matching to learn joint PDE coefficient-solution distributions, using guided sampling with composite losses for forward and inverse problems and providing error guarantees under stated assumptions.
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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.