Constrained Ulam Dynamic Mode Decomposition: Approximation of the Perron-Frobenius Operator for Deterministic and Stochastic Systems

Abstract

Dynamical systems described by ordinary and stochastic differential equations can be analyzed through the eigen-decomposition of the Perron–Frobenius (PF) and Koopman transfer operators. While the Koopman operator may be approximated by data-driven techniques, e.g., extended dynamic mode decomposition (EDMD), the approximation of the PF operator uses a single-pass Monte Carlo approach in Ulam’s method, which requires a sufficiently long time step. This letter proposes a finite-dimensional approximation technique for the PF operator that uses multi-pass Monte Carlo data to pose and solve a constrained EDMD-like least-squares problem to approximate the PF operator on a finite-dimensional basis. The basis functions used to project the PF operator are the characteristic functions of the state-space partitions. The results are analyzed theoretically and illustrated using deterministic and time-homogeneous stochastic systems.