langevin dynamics

Efficient sampling from unnormalized Boltzmann densities remains challenging, particularly in high-dimensional multimodal distributions and Bayesian inference. This work proposes a probability flow ordinary differential equation (ODE) method based on linear stochastic interpolation, which, for the first time, employs a Langevin sampler to jointly draw samples from interpolated distributions and estimate the corresponding velocity field. This approach provides a theoretically grounded initialization and dynamic modeling for the flow ODE, ensuring stability and convergence. By integrating stochastic interpolation, probability flow ODEs, and Langevin diffusion, the method demonstrates efficient and robust sampling performance across a range of high-dimensional multimodal distributions and Bayesian inference tasks.

This work addresses efficient sampling from high-dimensional, unnormalized target distributions. We propose a deterministic transport framework operating in finite time: an intermediate density flow is constructed via stochastic interpolation and modeled as a coupled forward-backward stochastic differential equation (FBSDE) system—the first application of FBSDEs to solve Hamilton–Jacobi–Bellman-type partial differential equations (PDEs)—enabling exact, finite-step density matching from a Gaussian initial distribution to the target. Unlike conventional diffusion models reliant on asymptotic iterative procedures, our approach bypasses such limitations by parameterizing the PDE solution via neural networks, yielding a differentiable and trainable transport map. Empirically, on high-dimensional multimodal and strongly correlated distributions, our method achieves 3–5× higher sampling efficiency than SGLD and DDPM, while significantly mitigating mode collapse and improving mode coverage.

Langevin dynamics sampling suffers from extremely low generation speed due to the requirement of numerous fine-grained iterative steps. To address this, we propose PID-controlled Langevin Dynamics (PIDLD), the first method to incorporate proportional-integral-derivative (PID) control into generative model sampling. PIDLD dynamically adjusts the step size using instantaneous energy gradient feedback (P), historical accumulation (I), and trend estimation of gradient changes (D), enabling adaptive and stable sampling trajectories without additional training or data. Our approach significantly reduces the number of sampling steps—by 40–60% on average—while simultaneously improving sample quality and convergence robustness. We validate PIDLD on image generation and inverse inference tasks, demonstrating consistent performance gains across diverse settings. This work establishes a novel paradigm for accelerating score-based generative models through principled control-theoretic design.

This work proposes a method to significantly improve sampling efficiency in generative models without requiring retraining, thereby drastically reducing the number of steps needed for generation. By introducing point-mass interpolation scheduling and a family of lazy schedulers, the approach unifies the sampling trajectories of flow models and diffusion models. Leveraging stochastic interpolation theory, SDE path transformations, and Gaussian–point-mass measure bridging, it enables seamless conversion of sample paths across different schedulers and diffusion coefficients. Notably, this is the first successful application of accelerated sampling from pretrained flow models to real-world non-Gaussian data, achieving substantial reductions in image generation steps while preserving high sample quality. The results demonstrate the theoretical soundness and practical efficacy of the proposed framework on complex datasets.

This work addresses the problem of efficiently and accurately bridging arbitrary probability density functions within a bounded time horizon. Methodologically, it introduces a unified generative modeling paradigm based on stochastic interpolation processes, seamlessly integrating flow-based and diffusion-based models—supporting both deterministic ordinary differential equation (ODE) paths and stochastic differential equation (SDE) paths with tunable noise. A novel score-matching objective is derived for the first time; theoretical analysis proves that optimizing only a quadratic loss suffices for likelihood control, overcoming the traditional limitation of deterministic models requiring additional Fisher divergence regularization. By unifying Schrödinger bridge theory, the Fokker–Planck equation, and variational inference, the framework rigorously recovers the Schrödinger bridge solution under optimal interpolation and provides a unified estimator for both likelihood and cross-entropy.

This work addresses the limitations of existing single-step generative models, which predominantly rely on ordinary differential equations and struggle to accurately model stochastic dynamical systems. The authors propose the Itô map—a theoretically grounded, exact flow mapping derived from Itô integral theory for stochastic differential equations—that enables one-step prediction of future states from intermediate states and Brownian paths with arbitrary step sizes. This approach supports end-to-end training and facilitates efficient, differentiable posterior sampling and stochastic control during inference. Experimental results demonstrate that the method generates diverse yet conditionally consistent terminal samples on both synthetic data and image generation tasks, significantly enhancing guided control capabilities.

This work proposes a unified variational generative modeling framework based on stochastic differential equations (SDEs) to efficiently address complex data generation tasks, including images, videos, and biomolecular structures. By incorporating both ordinary and stochastic differential equations, the authors derive the evidence lower bound (ELBO) from a variational inference perspective, systematically demonstrating that diffusion models, score matching, and flow matching are distinct parameterizations within this general framework. Through theoretical analysis grounded in the Fokker–Planck equation and empirical validation via one-dimensional density modeling experiments, the study provides clear comparisons among different parameterization strategies, confirming the proposed framework’s theoretical coherence, expressive capacity, and practical efficacy.

Diffusion posterior samplers are widely used in inverse problems, yet their outputs suffer from bias and discretization instability at low temperatures, with the underlying mechanisms poorly understood. This work constructs a tractable surrogate path bridging the true posterior and a standard Gaussian distribution, leveraging the Feynman–Kac formula to express the density ratio as a path-space expectation. For the first time, it derives a partial differential equation that characterizes sampling bias. By integrating Radon–Nikodym derivatives, Ornstein–Uhlenbeck processes, and auxiliary drift reconstruction, the study reveals the origins and spatial distribution of bias in methods such as DPS and STSL: it precisely identifies regions of over- and under-sampling in DPS and explains how STSL enhances stability through a smoothed reaction term. This theoretical framework provides a foundation for designing stable and efficient posterior sampling algorithms.

Diffusion-based generative models have achieved remarkable performance across various domains, yet their practical deployment is often limited by high sampling costs. While prior work focuses on training objectives or individual solvers, the holistic design of sampling, specifically solver selection and scheduling, remains dominated by static heuristics. In this work, we revisit this challenge through a geometric lens, proposing SDM, a principled framework that aligns the numerical solver with the intrinsic properties of the diffusion trajectory. By analyzing the ODE dynamics, we show that efficient low-order solvers suffice in early high-noise stages while higher-order solvers can be progressively deployed to handle the increasing non-linearity of later stages. Furthermore, we formalize the scheduling by introducing a Wasserstein-bounded optimization framework. This method systematically derives adaptive timesteps that explicitly bound the local discretization error, ensuring the sampling process remains faithful to the underlying continuous dynamics. Without requiring additional training or architectural modifications, SDM achieves state-of-the-art performance across standard benchmarks, including an FID of 1.93 on CIFAR-10, 2.41 on FFHQ, and 1.98 on AFHQv2, with a reduced number of function evaluations compared to existing samplers. Our code is available at https://github.com/aiimaginglab/sdm.

This work addresses the challenges of exploration and denoising stability in reinforcement learning post-training with flow matching models by introducing Precise Sampler—a stochastic sampling strategy consistent with stochastic differential equations (SDEs). The method approximates the clean latent posterior mean via a frozen estimate, thereby eliminating redundant noise inherent in standard discretization schemes and effectively balancing exploration and stability within a limited number of sampling steps. Integrating SDE scheduling, flow matching theory, and reinforcement learning, the proposed approach achieves state-of-the-art performance on benchmarks such as PickScore and HPSv2.1, while reducing training time by 13.1%–53.2%, significantly enhancing both the efficiency and convergence stability of reward optimization.