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。