Field dynamics inference via spectral density estimation

TL;DR

This paper introduces a method to infer the underlying laws of stochastic processes by estimating their spectral density from noisy, incomplete measurements, applicable to time-series and spatio-temporal data.

Contribution

It presents a novel approach to approximate stochastic process dynamics using spectral density estimation, extending inverse problem techniques with potential for broader application.

Findings

Effective reconstruction of stochastic dynamics from noisy data

Applicable to both time-series and spatio-temporal processes

Demonstrated on real-world measurement data

Abstract

Stochastic differential equations (SDEs) are of utmost importance in various scientific and industrial areas. They are the natural description of dynamical processes whose precise equations of motion are either not known or too expensive to solve, e.g., when modeling Brownian motion. In some cases, the equations governing the dynamics of a physical system on macroscopic scales occur to be unknown since they typically cannot be deduced from general principles. In this work, we describe how the underlying laws of a stochastic process can be approximated by the spectral density of the corresponding process. Furthermore, we show how the density can be inferred from possibly very noisy and incomplete measurements of the dynamical field. Generally, inverse problems like these can be tackled with the help of Information Field Theory (IFT). For now, we restrict to linear and autonomous processes. Though, this is a non-conceptual limitation that may be omitted in future work. To demonstrate its applicability we employ our reconstruction algorithm on a time-series and spatio-temporal processes.