Motivated by applications in queueing theory, we consider a stochastic
control problem whose state space is the $d$-dimensional positive orthant. The
controlled process $Z$ evolves as a reflected Brownian motion whose covariance
matrix is exogenously specified, as are its directions of reflection from the
orthant's boundary surfaces. A system manager chooses a drift vector
$\theta(t)$ at each time $t$ based on the history of $Z$, and the cost rate at
time $t$ depends on both $Z(t)$ and $\theta(t)$. In our initial problem
formulation, the objective is to minimize expected discounted cost over an
infinite planning horizon, after which we treat the corresponding ergodic
control problem. Extending earlier work by Han et al. (Proceedings of the
National Academy of Sciences, 2018, 8505-8510), we develop and illustrate a
simulation-based computational method that relies heavily on deep neural
network technology. For test problems studied thus far, our method is accurate
to within a fraction of one percent, and is computationally feasible in
dimensions up to at least $d=30$.

通过深度神经网络技术开发和展示了一种基于模拟的计算方法来解决随机控制问题，将其准确性扩展到 $d=30$ 的维度。

高维 RBM 的漂移控制：基于神经网络的计算方法

Drift Control of High-Dimensional RBM: A Computational Method Based on  Neural Networks

Denoising diffusion models have recently emerged as the predominant paradigm
for generative modelling. Their extension to Riemannian manifolds has
facilitated their application to an array of problems in the natural sciences.
Yet, in many practical settings, such manifolds are defined by a set of
constraints and are not covered by the existing (Riemannian) diffusion model
methodology. Recent work has attempted to address this issue by employing novel
noising processes based on logarithmic barrier methods or reflected Brownian
motions. However, the associated samplers are computationally burdensome as the
complexity of the constraints increases. In this paper, we introduce an
alternative simple noising scheme based on Metropolis sampling that affords
substantial gains in computational efficiency and empirical performance
compared to the earlier samplers. Of independent interest, we prove that this
new process corresponds to a valid discretisation of the reflected Brownian
motion. We demonstrate the scalability and flexibility of our approach on a
range of problem settings with convex and non-convex constraints, including
applications from geospatial modelling, robotics and protein design.

本文介绍了一种基于 Metropolis 抽样算法的简单去噪方案，证明了该新方法对应于反射布朗运动的有效离散化方法，并在包括地理空间建模、机器人技术和蛋白质设计在内的一系列问题设置上展示了我们方法的可扩展性和灵活性。