fokker-planck analysis

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.

To address the prevalent reward collapse problem in diffusion model fine-tuning, this paper proposes an entropy-regularized stochastic control framework and— for the first time—rigorously extends it to general *f*-divergence regularization. Methodologically, we formulate a continuous-time stochastic control model, integrating Itô calculus with variational inference to derive a computationally tractable and provably convergent optimal control policy. Theoretically, we establish that the proposed regularization effectively mitigates reward collapse; empirically, it significantly improves both sample quality and diversity. Key contributions include: (1) the first rigorous stochastic control analysis framework specifically designed for diffusion model fine-tuning; (2) a unified generalization of entropy regularization to arbitrary *f*-divergences, substantially enhancing methodological generality and robustness; and (3) a practical fine-tuning paradigm implementable under multiple divergence metrics.

This work addresses the lack of interpretability and resolution-invariant modeling capability of diffusion models in infinite-dimensional function spaces—such as images, time series, and probability density functions (PDFs). To this end, we extend stochastic optimal control theory to infinite dimensions. Our method establishes, for the first time, a rigorous equivalence between infinite-dimensional Doob h-transforms and stochastic optimal control; overcomes the fundamental challenge of undefined densities in infinite dimensions by constructing diffusion bridges without explicit density assumptions; and jointly leverages variational inference and functional optimization to directly compute optimal transport paths in function space. Experiments demonstrate that the proposed framework achieves resolution invariance in both distributional bridge learning and sampling tasks. It significantly improves fidelity and interpretability across diverse applications—including image generation, time-series interpolation, and PDF modeling—without dependence on spatial or temporal discretization.

This work investigates the fundamental speed–accuracy trade-off in diffusion models, establishing—for the first time—a theoretical connection to nonequilibrium stochastic thermodynamics. Methodologically, it links the entropy production rate (a kinetic speed measure in the absence of nonconservative forces) to generation accuracy, deriving a quantitative inequality between them; it further defines an “optimal learning protocol” via the 2-Wasserstein geodesic from optimal transport theory, revealing an intrinsic Pareto frontier between speed and fidelity. The approach integrates stochastic thermodynamics, Fokker–Planck analysis, Wasserstein geometry, and numerical diffusion modeling. Experiments across diverse noise schedules and real-world image datasets validate the trade-off, quantify distortion induced by nonconservative forces, and demonstrate that the optimal protocol significantly improves sampling efficiency and reconstruction quality.

To address the challenge of posterior sampling in high-dimensional, non-Gaussian Bayesian inverse problems where gradients are inaccessible, this paper proposes a gradient-free, affine-invariant ensemble sampling method. The core innovation couples second-order Langevin dynamics with Hamiltonian stochastic differential equations by introducing auxiliary momentum variables and designing a damping-driven mechanism, thereby constructing a novel stochastic dynamical system that preserves the target Gibbs measure. Furthermore, the method integrates covariance-adaptive preconditioning with ensemble averaging approximation to accelerate convergence without compromising invariance. This work establishes the first theoretical unification of second-order Langevin dynamics and ensemble approximation, significantly enhancing sampling efficiency and robustness. Extensive experiments on multiple high-dimensional Bayesian inverse problems demonstrate its superior performance over existing approaches.

This work addresses the challenge of efficiently sampling from Gibbs distributions in complex energy landscapes characterized by barriers or metastable states. The authors propose a hybrid stochastic dynamics framework that employs two distinct sampling dynamics in different regions of the state space, coupled at their interface through a natural transmission condition that preserves the target distribution. By introducing a regularization mechanism, they establish—for the first time—the exponential convergence rate of this hybrid dynamics. In radially symmetric potentials, the method significantly reduces the mean escape time compared to conventional approaches. Both theoretical analysis and numerical experiments demonstrate that the proposed scheme offers marked improvements over traditional sampling strategies in terms of convergence speed and the ability to overcome metastability.

This work addresses the challenge of simulating sample paths for stochastic differential equations (SDEs) with gradient drift and unit diffusion coefficients under noisy observations, where existing methods often suffer from discretization bias or high sampling complexity. The authors propose an exact Gibbs sampling framework that enables unbiased path simulation without temporal discretization and naturally integrates Gaussian process tools to facilitate parameter inference. This approach achieves, for the first time, discretization-free MCMC sampling for a broad class of SDE models, handling both univariate and multivariate cases within a unified framework—without requiring rejection sampling or debiasing techniques. Empirical evaluations on synthetic and real-world data demonstrate clear advantages over particle MCMC methods, offering superior accuracy and computational efficiency.

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.

This work addresses the limitations of existing stochastic interpolation methods, which are confined to finite-dimensional spaces and thus struggle to model generative tasks between arbitrary distributions in function spaces. For the first time, the authors extend stochastic interpolation theory to infinite-dimensional Hilbert spaces, establishing a rigorous mathematical framework grounded in functional analysis and stochastic differential equations. They provide well-posedness guarantees and explicit error bounds, thereby overcoming dimensional constraints and enabling controllable generation under complex conditions. The proposed method achieves state-of-the-art performance on PDE-driven function space benchmark tasks, offering a general and efficient tool for generating high-dimensional continuous distributions in scientific computing.

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.