Learning the Infinitesimal Generator of Stochastic Diffusion Processes
This work addresses a fundamental challenge in understanding numerical simulations of natural and physical systems, offering a novel method for a known bottleneck.
The paper tackles the problem of data-driven learning of the infinitesimal generator for stochastic diffusion processes, which is challenging due to its unbounded nature, by introducing a novel energy-based framework that integrates physical priors and provides dimension-independent learning bounds and non-spurious spectral estimation.
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.