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.
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.
A control-theoretic linear program yields value-driven transport policies for generative modeling with straight paths and simulation-free training.
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.
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.
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.
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.
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.
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.
DiffIML applies score-based generative modeling to image manipulation localization, recovering coherent masks iteratively from noise to improve generalization on unseen manipulation types.
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.
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.
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.
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.
-
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.
-
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.
-
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.
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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.
-
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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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.
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Muninn: Your Trajectory Diffusion Model But Faster
Muninn accelerates diffusion trajectory planners up to 4.6x by spending an uncertainty budget to decide when to cache denoiser outputs, preserving performance and certifying bounded deviation from full computation.
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Discrete Langevin-Inspired Posterior Sampling
ΔLPS is a gradient-guided discrete posterior sampler for inverse problems that works with masked or uniform discrete diffusion priors and outperforms prior discrete methods on image restoration tasks.
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Remix the Timbre: Diffusion-Based Style Transfer Across Polyphonic Stems
MixtureTT performs direct per-stem timbre transfer on polyphonic mixtures via a shared diffusion transformer, outperforming single-stem baselines on SATB choral data while eliminating cascaded separation errors.
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A Call to Lagrangian Action: Learning Population Mechanics from Temporal Snapshots
Wasserstein Lagrangian Mechanics formalizes second-order dynamics in Wasserstein space and provides an algorithm to learn them from observed marginals without specifying the Lagrangian, outperforming gradient flows on various dynamics.
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Adaptive Subspace Projection for Generative Personalization
A training-free adaptive subspace projection method mitigates semantic collapsing in generative personalization by isolating and adjusting drift in a low-dimensional subspace using the stable pre-trained embedding as anchor.
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Kurtosis-Guided Denoising Score Matching for Tabular Anomaly Detection
K-DSM uses per-feature kurtosis to set noise scales in DSM, enabling effective single-scale anomaly detection on tabular benchmarks in both semi-supervised and unsupervised settings.
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Arena as Offline Reward: Efficient Fine-Grained Preference Optimization for Diffusion Models
ArenaPO infers Gaussian capability distributions from pairwise preferences and applies truncated-normal latent inference to derive fine-grained offline rewards for preference optimization of text-to-image diffusion models.
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DBMSolver: A Training-free Diffusion Bridge Sampler for High-Quality Image-to-Image Translation
DBMSolver is a new training-free sampler using exponential integrators that reduces NFEs by up to 5x and improves quality in diffusion bridge model-based image-to-image translation tasks.
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Beyond Penalization: Diffusion-based Out-of-Distribution Detection and Selective Regularization in Offline Reinforcement Learning
DOSER detects OOD actions via diffusion-model denoising error and applies selective regularization based on predicted transitions, proving gamma-contraction with performance bounds and outperforming priors on offline RL benchmarks.
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Tempered Guided Diffusion
Tempered Guided Diffusion uses annealed SMC to produce consistent particle approximations to the posterior for training-free conditional diffusion sampling, outperforming independent guided trajectories in experiments.
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PODiff: Latent Diffusion in Proper Orthogonal Decomposition Space for Scientific Super-Resolution
PODiff performs conditional diffusion in a fixed, variance-ordered POD latent space to enable efficient probabilistic super-resolution of high-dimensional scientific fields with lower memory and better-calibrated uncertainty than pixel-space or dropout baselines.
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Inferring Active Neural Circuits Using Diffusion Scores
SBTG recovers the Jacobian of the nonlinear transition map between brain states by multiplying cross-block scores from denoising models, enabling inference of lag-specific directed interactions in neural population data such as C. elegans calcium imaging.
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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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Decentralized Proximal Stochastic Gradient Langevin Dynamics
DE-PSGLD is the first decentralized MCMC sampler for constrained convex domains that converges to a regularized Gibbs distribution with explicit 2-Wasserstein bounds for agents and network averages.
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GD4: Graph-based Discrete Denoising Diffusion for MIMO Detection
GD4 is a graph-based discrete denoising diffusion method for MIMO detection that yields higher-quality suboptimal solutions than prior diffusion detectors and classical baselines under similar compute budgets in both under- and over-determined settings.
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ABC: Any-Subset Autoregression via Non-Markovian Diffusion Bridges in Continuous Time and Space
ABC enables any-subset autoregressive generation of continuous stochastic processes via non-Markovian diffusion bridges that track physical time and allow path-dependent conditioning.
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Oracle Noise: Faster Semantic Spherical Alignment for Interpretable Latent Optimization
Oracle Noise optimizes diffusion model noise on a Riemannian hypersphere guided by key prompt words to preserve the Gaussian prior, eliminate norm inflation, and achieve faster semantic alignment than Euclidean methods.
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$Z^2$-Sampling: Zero-Cost Zigzag Trajectories for Semantic Alignment in Diffusion Models
Z²-Sampling implicitly realizes zero-cost zigzag trajectories for curvature-aware semantic alignment in diffusion models by reducing multi-step paths via operator dualities and temporal caching while synthesizing a directional derivative penalty.
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Dream-Cubed: Controllable Generative Modeling in Minecraft by Training on Billions of Cubes
Dream-Cubed releases a billion-scale voxel dataset and 3D diffusion models that generate controllable Minecraft worlds by operating directly on blocks.
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Frequency-Forcing: From Scaling-as-Time to Soft Frequency Guidance
Frequency-Forcing guides pixel flow-matching with a data-derived low-frequency auxiliary stream to softly enforce scale-ordered generation, improving FID on ImageNet-256 over baselines.
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HP-Edit: A Human-Preference Post-Training Framework for Image Editing
HP-Edit introduces a post-training framework and RealPref-50K dataset that uses a VLM-based HP-Scorer to align diffusion image editing models with human preferences, improving outputs on Qwen-Image-Edit-2509.