We address data-driven learning of the infinitesimal generator of stochastic
diffusion processes, essential for understanding numerical simulations of
natural and physical systems. The unbounded nature of the generator poses
significant challenges, rendering conventional analysis techniques for
Hilbert-Schmidt operators ineffective. To overcome this, we introduce a novel
framework based on the energy functional for these stochastic processes. Our
approach integrates physical priors through an energy-based risk metric in both
full and partial knowledge settings. We evaluate the statistical performance of
a reduced-rank estimator in reproducing kernel Hilbert spaces (RKHS) in the
partial knowledge setting. Notably, our approach provides learning bounds
independent of the state space dimension and ensures non-spurious spectral
estimation. Additionally, we elucidate how the distortion between the intrinsic
energy-induced metric of the stochastic diffusion and the RKHS metric used for
generator estimation impacts the spectral learning bounds.

我们提出了一个基于能量泛函的新框架，用于学习随机扩散过程的微小生成器，该方法在完全和部分知识设置下通过能量风险度量整合物理先验，并通过减少秩估计器在重构核希尔伯特空间中评估统计性能，不受状态空间维度限制的限制，保证了非虚假谱估计，并阐明了随机扩散的内在能量引入度量与生成器估计中使用的重构核希尔伯特空间度量之间的失真对谱学习边界的影响。

学习随机扩散过程的微生成器

Learning the Infinitesimal Generator of Stochastic Diffusion Processes

In this paper we develop a Model Predictive Path Integral (MPPI) control
algorithm based on a generalized importance sampling scheme and perform
parallel optimization via sampling using a Graphics Processing Unit (GPU). The
proposed generalized importance sampling scheme allows for changes in the drift
and diffusion terms of stochastic diffusion processes and plays a significant
role in the performance of the model predictive control algorithm. We compare
the proposed algorithm in simulation with a model predictive control version of
differential dynamic programming.

本文发展了一种基于广义重要性抽样方案的模型预测路径积分 (MPPI) 控制算法，并使用图形处理器 (GPU) 进行均衡优化。该广义重要性抽样方案允许在随机扩散过程的漂移和扩散项中进行变化，并在模型预测控制算法的性能中发挥重要作用。本研究在仿真中将所提出的算法与微分动态规划的模型预测控制版本进行了比较。