fokker-planck control

This work addresses the coupled challenge of spatial safety and energy sustainability in long-term operation of multi-robot systems. The authors propose a density evolution model based on the Fokker–Planck partial differential equation, which uniquely integrates PDE-constrained optimization with control Lyapunov functions and control barrier functions. This integration simultaneously ensures collision avoidance, accurate tracking of desired spatial densities, and satisfaction of energy sufficiency constraints across multiple charging cycles. An efficient online control strategy is realized through real-time quadratic programming. The approach is validated through both physical experiments and large-scale simulations, demonstrating its effectiveness under localization and motion uncertainties. The method enables provably safe, energy-sustainable, and stable long-term autonomous operation of multi-robot systems.

Existing continuous-time diffusion models rely on pre-specified optimal processes or expensive numerical simulations, limiting adaptability to general objective functions. This paper proposes a simulation-free joint modeling framework that unifies time-varying density functions and diffusion dynamics via shared neural parameterization, directly enforcing the Fokker–Planck equation and probability conservation. Its core innovation is coupled neural parameterization—extending and simplifying the neural continuity equation—to enable end-to-end joint optimization of density evolution and drift-diffusion dynamics without numerical integration or sampling. The method intrinsically embeds physical constraints, supporting diverse tasks including generative modeling, dynamic optimal transport, and stochastic optimal control. Experiments on spatiotemporal event modeling and collective dynamics learning demonstrate high accuracy and strong generalization, validating both theoretical soundness and practical efficacy.

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 adaptive optimal control of linear diffusion processes—governed by linear stochastic differential equations—with unknown drift matrices, a setting prevalent in continuous-decision domains such as medical intervention and flight control where system dynamics are uncertain. We propose the first method embedding Thompson sampling into a continuous-time stochastic optimal control framework. Theoretically, it achieves an $ ilde{O}(sqrt{T})$ cumulative regret bound and guarantees short-term system stability. A key conceptual innovation is the introduction of the “optimality manifold,” a geometric construct enabling local analysis of parameter sensitivity and stability boundaries. Extensive simulations demonstrate superior performance over baseline algorithms in aircraft attitude regulation and artificial pancreas blood glucose control tasks, particularly improving worst-case regret.

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 theoretical grounding in noise scheduling for diffusion models, which has hindered the understanding of empirically effective scheduling strategies. The authors formulate the problem for the first time as an optimal control problem, where the Fisher information serves as the state variable and the noise schedule acts as the control input, with the objective of minimizing an upper bound on the KL sampling error. Within this framework, they derive sufficient conditions for achieving near-optimal sampling error and obtain a tunable closed-form expression for the noise schedule. The proposed method unifies and generalizes exponential and sigmoidal schedules, and, after parameter tuning, achieves improved Fréchet Inception Distance (FID) scores on standard image generation benchmarks.

This study addresses the limitations of traditional proportional wealth taxation, which merely shifts the drift term of wealth dynamics without altering the shape of the stationary distribution. The authors formulate optimal wealth redistribution as a control problem within the Fokker–Planck equation framework, introducing progressive taxation as a confining potential and transfer payments as source-sink terms to break drift symmetry—thereby enabling active shaping of the steady-state wealth distribution for the first time. Their approach integrates stochastic control, spectral gap analysis, and a McKean–Vlasov general equilibrium setting, naturally incorporating tax evasion costs and diminishing returns. Theoretical results yield a closed-form expression for the Gini coefficient under progressive taxation, demonstrating its capacity to significantly reduce inequality on policy-relevant timescales, in stark contrast to proportional taxation, which relies inefficiently on slow demographic turnover.

研究了未知漂移系数和运行奖励函数的随机控制问题，通过强化学习框架开发出有效的算法来学习最优值函数和控制策略。

Existing diffusion models struggle to achieve effective and theoretically guaranteed control during inference when guided by distribution-level rewards, such as diversity or population statistics. This work addresses this challenge by formulating it as the approximation of a tilted measure within a mean-field framework and introduces a weighted interacting particle scheme that, for the first time, provides theoretical guarantees for distribution-level reward control. The proposed approach unifies existing pointwise reward control paradigms and establishes a theoretical foundation for batch guidance strategies. Empirical results demonstrate that the method accurately approximates target distributions in low-dimensional analytically tractable tasks and achieves superior performance in high-dimensional protein conformation generation.

In this article, we study unbalanced optimal transport (UOT) and establish a control-theoretic dynamical extension, which we call the unbalanced density control (UDC), for a class of Gaussian reference measures. In the static setting, we consider UOT with quadratic transport cost and Kullback--Leibler penalties on the marginals relative to prescribed Gaussian measures. We show that the infinite-dimensional variational problem admits an exact Gaussian reduction, yielding a finite-dimensional optimization over masses, means, and covariances, together with a closed-form expression for the optimal transported mass. We then formulate UDC for discrete-time linear systems, where the initial and terminal state measures are imposed softly through KL penalties and the intermediate evolution is governed by controlled linear dynamics with quadratic control cost. For this problem, we prove that any feasible solution can be replaced, without loss of optimality, by a Gaussian initial measure and an affine-Gaussian control policy. This leads to an exact finite-dimensional reformulation and, after a standard covariance-steering lifting, to an SDP-based optimization for fixed mass, again coupled with a closed-form mass update. We further establish existence of optimal solutions and identify a sufficient condition under which the affine-Gaussian UDC policy is deterministic. These results provide globally optimal solution methods for both Gaussian UOT and Gaussian UDC. Finally, we illustrate our results with several numerical examples.