Learning the Infinitesimal Generator of Stochastic Diffusion Processes

TL;DR

This paper presents a new framework for data-driven learning of the infinitesimal generator of stochastic diffusion processes, overcoming challenges posed by unbounded operators and incorporating physical priors for improved spectral estimation.

Contribution

It introduces an energy functional-based approach that handles unbounded generators, provides dimension-independent learning bounds, and analyzes the impact of energy-induced metrics on spectral estimation.

Findings

Provides a novel energy-based framework for generator learning.

Achieves spectral estimation without spurious results.

Offers dimension-independent statistical performance bounds.

Abstract

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.