Data-Driven Approximation of the Perron-Frobenius Operator Using the Wasserstein Metric

TL;DR

This paper presents a novel data-driven method for approximating the Perron-Frobenius Operator using distributional data and the Wasserstein metric, enabling better modeling of distributional dynamics.

Contribution

It introduces a new regression-based approach leveraging the Wasserstein metric for approximating the Perron-Frobenius Operator from distributional snapshots.

Findings

The method effectively interpolates distributional flows.

Numerical simulations demonstrate the approach's viability.

The framework provides a gradient flow approximation algorithm.

Abstract

This manuscript introduces a regression-type formulation for approximating the Perron-Frobenius Operator by relying on distributional snapshots of data. These snapshots may represent densities of particles. The Wasserstein metric is leveraged to define a suitable functional optimization in the space of distributions. The formulation allows seeking suitable dynamics so as to interpolate the distributional flow in function space. A first-order necessary condition for optimality is derived and utilized to construct a gradient flow approximating algorithm. The framework is exemplied with numerical simulations.