Nonparametric Uncertainty Quantification for Stochastic Gradient Flows

TL;DR

This paper introduces a nonparametric, data-driven approach using diffusion maps to quantify uncertainty in stochastic gradient flows, enabling model-free solutions to prediction, filtering, and response problems.

Contribution

It develops a novel nonparametric method leveraging diffusion maps to approximate the Kolmogorov operator for uncertainty quantification in stochastic gradient systems.

Findings

Successfully applied to linear gradient flow with known solutions

Extended to chaotic nonlinear gradient flow systems

Demonstrated effectiveness without explicit model assumptions

Abstract

This paper presents a nonparametric statistical modeling method for quantifying uncertainty in stochastic gradient systems with isotropic diffusion. The central idea is to apply the diffusion maps algorithm to a training data set to produce a stochastic matrix whose generator is a discrete approximation to the backward Kolmogorov operator of the underlying dynamics. The eigenvectors of this stochastic matrix, which we will refer to as the diffusion coordinates, are discrete approximations to the eigenfunctions of the Kolmogorov operator and form an orthonormal basis for functions defined on the data set. Using this basis, we consider the projection of three uncertainty quantification (UQ) problems (prediction, filtering, and response) into the diffusion coordinates. In these coordinates, the nonlinear prediction and response problems reduce to solving systems of infinite-dimensional linear ordinary differential equations. Similarly, the continuous-time nonlinear filtering problem reduces to solving a system of infinite-dimensional linear stochastic differential equations. Solving the UQ problems then reduces to solving the corresponding truncated linear systems in finitely many diffusion coordinates. By solving these systems we give a model-free algorithm for UQ on gradient flow systems with isotropic diffusion. We numerically verify these algorithms on a 1-dimensional linear gradient flow system where the analytic solutions of the UQ problems are known. We also apply the algorithm to a chaotically forced nonlinear gradient flow system which is known to be well approximated as a stochastically forced gradient flow.