Flux Matching generalizes score-based generative modeling by using a weaker objective that admits infinitely many non-conservative vector fields with the data as stationary distribution, enabling new design choices beyond traditional score matching.
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Score-Based Generative Modeling through Stochastic Differential Equations
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abstract
Creating noise from data is easy; creating data from noise is generative modeling. We present a stochastic differential equation (SDE) that smoothly transforms a complex data distribution to a known prior distribution by slowly injecting noise, and a corresponding reverse-time SDE that transforms the prior distribution back into the data distribution by slowly removing the noise. Crucially, the reverse-time SDE depends only on the time-dependent gradient field (\aka, score) of the perturbed data distribution. By leveraging advances in score-based generative modeling, we can accurately estimate these scores with neural networks, and use numerical SDE solvers to generate samples. We show that this framework encapsulates previous approaches in score-based generative modeling and diffusion probabilistic modeling, allowing for new sampling procedures and new modeling capabilities. In particular, we introduce a predictor-corrector framework to correct errors in the evolution of the discretized reverse-time SDE. We also derive an equivalent neural ODE that samples from the same distribution as the SDE, but additionally enables exact likelihood computation, and improved sampling efficiency. In addition, we provide a new way to solve inverse problems with score-based models, as demonstrated with experiments on class-conditional generation, image inpainting, and colorization. Combined with multiple architectural improvements, we achieve record-breaking performance for unconditional image generation on CIFAR-10 with an Inception score of 9.89 and FID of 2.20, a competitive likelihood of 2.99 bits/dim, and demonstrate high fidelity generation of 1024 x 1024 images for the first time from a score-based generative model.
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- abstract Creating noise from data is easy; creating data from noise is generative modeling. We present a stochastic differential equation (SDE) that smoothly transforms a complex data distribution to a known prior distribution by slowly injecting noise, and a corresponding reverse-time SDE that transforms the prior distribution back into the data distribution by slowly removing the noise. Crucially, the reverse-time SDE depends only on the time-dependent gradient field (\aka, score) of the perturbed data distribution. By leveraging advances in score-based generative modeling, we can accurately estimate
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representative citing papers
A-CODE presents a fully atomic one-stage multimodal diffusion model for protein co-design that claims superior unconditional generation performance over prior one- and two-stage models plus a tenfold success-rate gain on hard binder-design tasks.
FMRG reformulates guidance as deterministic optimal control, deriving a single-trajectory method using the flow map that matches or exceeds baselines on reward-guided generation and inverse problems with 3 NFEs at text-to-image scale.
Quotient-space diffusion models generate correct symmetric distributions by removing redundancy on the quotient space, simplifying learning and improving results on small molecules and proteins under SE(3) symmetry.
The García-Pintos feedback Hamiltonian equals the score function of the quantum trajectory distribution, linking quantum feedback to diffusion-model reversal.
Diffusion sampling from d-dimensional distributions requires at least ~sqrt(d) adaptive score queries when score estimates have polynomial accuracy.
OP-GRPO is the first off-policy GRPO method for flow-matching models that reuses trajectories via replay buffer and importance sampling corrections, matching on-policy performance with 34.2% of the training steps.
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.
ASTRA reframes transition-state search as guided diffusion inference that samples the isodensity surface between metastable basins and converges to first-order saddles via score differences and physical forces.
MF-PID turns independent diffusion samples into mean-field interacting agents, proving that quadratic interactions yield exact linear mean interpolation and delivering 19-24% energy savings in demand-response control.
Föllmer processes are variationally optimal among generative diffusions because they minimize the impact of drift estimation error on path-space KL divergence, rendering different interpolation schedules statistically equivalent.
Flow-GRPO is the first online RL method for flow matching models, raising GenEval accuracy from 63% to 95% and text-rendering accuracy from 59% to 92% with little reward hacking.
LLaDA is a scalable diffusion-based language model that matches autoregressive LLMs like LLaMA3 8B on tasks and surpasses GPT-4o on reversal poem completion.
DDIMs construct non-Markovian diffusion processes that share DDPM training objectives but allow much faster reverse sampling, demonstrated empirically at 10-50x wall-clock speedup.
Introduces a Bridge latent interface that maps mismatched student latents into teacher space, enabling distillation from modern diffusion teachers to compact one-step students and raising SD 1.5 HPSv3 from 5.4 to 9.4 while keeping one-step speed.
LA-SR redefines unpaired super-resolution in language space by projecting images into a semantically rich representation and applying vision-language model guided losses to handle real-world degradations extracted from depth variations.
FlexiSLM is the first spoken language model supporting dynamic and controllable frame rates on speech input and output, outperforming fixed-rate 7B models at high quality and enabling faster inference at lower rates like 6.25 Hz.
ReMatch corrects train-test residual distribution mismatch in probabilistic downscaling via optimal transport in low-dimensional PCA space, reducing under-dispersion and improving SSR and CRPS on HRRR-ERA5 wind data.
Hybrid simulation and non-Euclidean elasticity theory demonstrate that clathrin coats develop adaptive rigidity and memory during growth, producing flat, stalled, or closed outcomes through two energy-landscape gates and matching experiments without fitted parameters.
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.
CORDEX-ML-Bench benchmarks 40 ML models for climate downscaling and finds generative models outperform deterministic ones on precipitation while historically trained models underestimate future climate signals.
QMC applied to Euler-Maruyama yields faster sampling-error decay than Monte Carlo, and the new MSTG method based on exact simulation achieves super-exponential truncation-error decay that sharply reduces integration dimension.
STREAM decouples text and music conditioning in a diffusion transformer via AdaLN for structure and BEAM for beats, plus new Motorica++ dataset and editability metrics, claiming SOTA music alignment with preserved semantics.
Direct fixed-weight solver for free-support Wasserstein medians relocates atoms using OT barycentric projections and inverse-distance weights, achieving monotone descent on smoothed objectives with fewer subproblems than nested Weiszfeld baselines.
citing papers explorer
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Generative Modeling with Flux Matching
Flux Matching generalizes score-based generative modeling by using a weaker objective that admits infinitely many non-conservative vector fields with the data as stationary distribution, enabling new design choices beyond traditional score matching.
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A-CODE: Fully Atomic Protein Co-Design with Unified Multimodal Diffusion
A-CODE presents a fully atomic one-stage multimodal diffusion model for protein co-design that claims superior unconditional generation performance over prior one- and two-stage models plus a tenfold success-rate gain on hard binder-design tasks.
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How to Guide Your Flow: Few-Step Alignment via Flow Map Reward Guidance
FMRG reformulates guidance as deterministic optimal control, deriving a single-trajectory method using the flow map that matches or exceeds baselines on reward-guided generation and inverse problems with 3 NFEs at text-to-image scale.
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Quotient-Space Diffusion Models
Quotient-space diffusion models generate correct symmetric distributions by removing redundancy on the quotient space, simplifying learning and improving results on small molecules and proteins under SE(3) symmetry.
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The Feedback Hamiltonian is the Score Function: A Diffusion-Model Framework for Quantum Trajectory Reversal
The García-Pintos feedback Hamiltonian equals the score function of the quantum trajectory distribution, linking quantum feedback to diffusion-model reversal.
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Query Lower Bounds for Diffusion Sampling
Diffusion sampling from d-dimensional distributions requires at least ~sqrt(d) adaptive score queries when score estimates have polynomial accuracy.
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OP-GRPO: Efficient Off-Policy GRPO for Flow-Matching Models
OP-GRPO is the first off-policy GRPO method for flow-matching models that reuses trajectories via replay buffer and importance sampling corrections, matching on-policy performance with 34.2% of the training steps.
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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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A Priori Sampling of Transition States with Guided Diffusion
ASTRA reframes transition-state search as guided diffusion inference that samples the isodensity surface between metastable basins and converges to first-order saddles via score differences and physical forces.
-
Mean-Field Path-Integral Diffusion: From Samples to Interacting Agents
MF-PID turns independent diffusion samples into mean-field interacting agents, proving that quadratic interactions yield exact linear mean interpolation and delivering 19-24% energy savings in demand-response control.
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Variational Optimality of F\"ollmer Processes in Generative Diffusions
Föllmer processes are variationally optimal among generative diffusions because they minimize the impact of drift estimation error on path-space KL divergence, rendering different interpolation schedules statistically equivalent.
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Flow-GRPO: Training Flow Matching Models via Online RL
Flow-GRPO is the first online RL method for flow matching models, raising GenEval accuracy from 63% to 95% and text-rendering accuracy from 59% to 92% with little reward hacking.
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Large Language Diffusion Models
LLaDA is a scalable diffusion-based language model that matches autoregressive LLMs like LLaMA3 8B on tasks and surpasses GPT-4o on reversal poem completion.
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Denoising Diffusion Implicit Models
DDIMs construct non-Markovian diffusion processes that share DDPM training objectives but allow much faster reverse sampling, demonstrated empirically at 10-50x wall-clock speedup.
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Cross-Space Distillation: Teaching One-Step Students with Modern Diffusion Teachers
Introduces a Bridge latent interface that maps mismatched student latents into teacher space, enabling distillation from modern diffusion teachers to compact one-step students and raising SD 1.5 HPSv3 from 5.4 to 9.4 while keeping one-step speed.
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Language-Assisted Super-Resolution from Real-World Low-Resolution Patches
LA-SR redefines unpaired super-resolution in language space by projecting images into a semantically rich representation and applying vision-language model guided losses to handle real-world degradations extracted from depth variations.
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FlexiSLM: A Dynamic and Controllable Frame Rate Spoken Language Model
FlexiSLM is the first spoken language model supporting dynamic and controllable frame rates on speech input and output, outperforming fixed-rate 7B models at high quality and enabling faster inference at lower rates like 6.25 Hz.
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Mind the Residual Gap: Probabilistic Downscaling under Real-World Bias
ReMatch corrects train-test residual distribution mismatch in probabilistic downscaling via optimal transport in low-dimensional PCA space, reducing under-dispersion and improving SSR and CRPS on HRRR-ERA5 wind data.
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Pathway variability, coat stiffening and mechanical adaptation during clathrin-mediated endocytosis
Hybrid simulation and non-Euclidean elasticity theory demonstrate that clathrin coats develop adaptive rigidity and memory during growth, producing flat, stalled, or closed outcomes through two energy-landscape gates and matching experiments without fitted parameters.
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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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CORDEX-ML-Bench: A Benchmark for Data-Driven Regional Climate Downscaling -Experiment Design and Overview
CORDEX-ML-Bench benchmarks 40 ML models for climate downscaling and finds generative models outperform deterministic ones on precipitation while historically trained models underestimate future climate signals.
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Quasi-Monte Carlo for SDE Simulation: Error Analysis and Dimensionality Reduction
QMC applied to Euler-Maruyama yields faster sampling-error decay than Monte Carlo, and the new MSTG method based on exact simulation achieves super-exponential truncation-error decay that sharply reduces integration dimension.
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Text Dictates, Music Decorates: Energy-based Attention for Editable Dance Motion Generation
STREAM decouples text and music conditioning in a diffusion transformer via AdaLN for structure and BEAM for beats, plus new Motorica++ dataset and editability metrics, claiming SOTA music alignment with preserved semantics.
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Fast Computation of Free-Support Wasserstein Medians
Direct fixed-weight solver for free-support Wasserstein medians relocates atoms using OT barycentric projections and inverse-distance weights, achieving monotone descent on smoothed objectives with fewer subproblems than nested Weiszfeld baselines.
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Chameleon: Style-Content Disentangled Framework for Cross-Domain Object Compositing
Chameleon proposes the first large-scale cross-domain compositing dataset and a disentangled encoder plus gated diffusion transformer that outperforms prior in-domain and cross-domain methods on plausibility and fidelity.
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YoCausal: How Far is Video Generation from World Model? A Causality Perspective
YoCausal benchmark shows video diffusion models detect the arrow of time but lack genuine causal understanding relative to humans.
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Sample-Efficient Diffusion-based Reinforcement Learning with Critic Guidance
CGPO integrates training-free critic guidance into diffusion denoising to produce high-Q actions as regression targets, yielding SOTA results on MuJoCo locomotion and successful Franka arm grasping.
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Spectral Guidance for Flexible and Efficient Control of Diffusion Models
Spectral Guidance learns singular functions via self-supervised objective to project guidance signals onto diffusion sampling trajectories, enabling stable control without retraining or backpropagation and improving CIFAR-10 accuracy by 37 points with 4x faster sampling.
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Parameter-Efficient Generative Modeling with Controlled Vector Fields
Presents a controlled vector field framework for continuous generative modeling where velocity is formed from fixed bracket-generating fields modulated by scalar controls, with an expressivity principle under controllability assumptions.
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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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Simulation-Informed Diffusion for Decentralized Multi-robot Motion Planning
SID is a decentralized planner that reuses a constraint-aware diffusion model to simulate neighbors' trajectories and then generate collision-free own paths, enabling minimal communication and scaling to 108 robots.
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Everything at Every Scale: Scale-Invariant Diffusion with Continuous Super-Resolution
SKILD unifies unconditional image generation and continuous super-resolution in one diffusion model via scale-invariant k-space dynamics where the reverse process handles both tasks by varying only the starting timestep.
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Towards Anatomically Plausible Human Image Generation via Synthetic Localized Preferences
ASAP generates over 10K synthetic anatomical preference pairs via targeted degradation of high-fidelity images and applies a localized margin-bounded DPO to reduce anatomical errors in text-to-image human generation, supported by the new HAP dataset and HAF-Bench.
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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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Let EEG Models Learn EEG
JET is a conditional flow matching framework that generates EEG as continuous raw sequences with added constraints for spectral and temporal properties, achieving over 40% lower TS-FID than prior discrete denoising methods on three benchmarks.
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Linear-DPO: Linear Direct Preference Optimization for Diffusion and Flow-Matching Generative Models
Linear-DPO replaces sigmoid utility with linear utility and adds EMA reference to improve preference alignment in diffusion and flow-matching text-to-image models.
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CAdam: Context-Adaptive Moment Estimation for 3D Gaussian Densification in Generative Distillation
CAdam reinterprets densification in generative 3DGS as signal verification via gradient-moment interference, quantile context, and SNR gating to achieve large reductions in primitive count with comparable quality.
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Mat\'ern Noise for Triangulation-Agnostic Flow Matching on Meshes
Proposes discretized Matérn process noise for triangulation-agnostic flow matching on meshes with PoissonNet denoiser, tested on elastic states and humanoid poses for meshes exceeding one million triangles.
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Nested-GPT for variable-multiplicity parton showers: A case study in the resummation of non-global logarithms
Nested-GPT is an autoregressive Transformer surrogate that generates variable-multiplicity parton showers while enforcing ordered Markovian branching and matches reference Monte Carlo results for leading-log non-global logarithm resummation in the large-Nc limit.
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Functionalization via Structure Completion and Motion Rectification
Object functionalization is cast as neural graph completion over a functional graph of parts, contacts, and motions, followed by geometry realization that also rectifies erroneous motions, demonstrated on furniture with a new paired dataset.
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Towards Generalized Image Manipulation Localization via Score-based Model
DiffIML applies score-based generative modeling to image manipulation localization, recovering coherent masks iteratively from noise to improve generalization on unseen manipulation types.
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Training-Free Generative Sampling via Moment-Matched Score Smoothing
MM-SOLD is a training-free particle sampler whose large-particle limit converges to a moment-matched Gibbs distribution obtained by exponentially tilting a score-smoothed target.
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Sampling from Flow Language Models via Marginal-Conditioned Bridges
Marginal-conditioned bridges enable training-free sampling from Flow Language Models by drawing clean one-hot endpoints from factorized posteriors and using Ornstein-Uhlenbeck bridges, preserving token marginals and reducing denoising error versus conditional-mean bridges.
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HIR-ALIGN: Enhancing Hyperspectral Image Restoration via Diffusion-Based Data Generation
HIR-ALIGN augments limited target data for hyperspectral restoration by creating proxy clean images, synthesizing aligned HSIs with blur-robust diffusion and warp-based transfer, then finetuning models to lower target-domain risk.
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Proximal-Based Generative Modeling for Bayesian Inverse Problems
PGM framework links diffusion to proximal regularization for closed-form Moreau-score sampling in Bayesian inverse problems, learned only from prior samples.
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Edit-Compass & EditReward-Compass: A Unified Benchmark for Image Editing and Reward Modeling
Edit-Compass and EditReward-Compass are new unified benchmarks for fine-grained image editing evaluation and realistic reward modeling in reinforcement learning optimization.
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Bridging Domain Gaps with Target-Aligned Generation for Offline Reinforcement Learning
TCE bridges domain gaps in offline RL by selectively using source data or generating target-aligned transitions via a dual score-based model, outperforming baselines in experiments.
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Amortized Guidance for Image Inpainting with Pretrained Diffusion Models
AID amortizes guidance for diffusion inpainting by training a reusable module via an auxiliary Gaussian formulation and continuous-time actor-critic algorithm, improving quality-speed trade-off with under 1% overhead.
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Aligning Flow Map Policies with Optimal Q-Guidance
Flow map policies enable fast one-step inference for flow-based RL policies, and FMQ provides an optimal closed-form Q-guided target for offline-to-online adaptation under trust-region constraints, achieving SOTA performance.
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On the Approximation Complexity of Matrix Product Operator Born Machines
MPO-BMs have NP-hard KL approximation in continuous settings but admit efficient polynomial-bond-dimension approximations with provable KL guarantees for structured targets under locality and spectral-gap conditions.