# On Convergence of Extended Dynamic Mode Decomposition to the Koopman Operator

## TL;DR

This paper proves the convergence of Extended Dynamic Mode Decomposition (EDMD) to the Koopman operator as the number of observables increases, ensuring reliable predictions and spectral analysis for dynamical systems.

## Contribution

It establishes the convergence of EDMD to the Koopman operator in the limit of increasing observables and proposes an analytic version of EDMD for direct construction without sampling.

## Key findings

- EDMD converges to the Koopman operator as N approaches infinity.
- Spectral properties of EDMD approximate those of the Koopman operator.
- The analytic EDMD method enables direct computation of the operator without sampling.

## Abstract

Extended Dynamic Mode Decomposition (EDMD) is an algorithm that approximates the action of the Koopman operator on an $N$-dimensional subspace of the space of observables by sampling at $M$ points in the state space. Assuming that the samples are drawn either independently or ergodically from some measure $\mu$, it was shown that, in the limit as $M\rightarrow\infty$, the EDMD operator $\mathcal{K}_{N,M}$ converges to $\mathcal{K}_N$, where $\mathcal{K}_N$ is the $L_2(\mu)$-orthogonal projection of the action of the Koopman operator on the finite-dimensional subspace of observables. In this work, we show that, as $N \rightarrow \infty$, the operator $\mathcal{K}_N$ converges in the strong operator topology to the Koopman operator. This in particular implies convergence of the predictions of future values of a given observable over any finite time horizon, a fact important for practical applications such as forecasting, estimation and control. In addition, we show that accumulation points of the spectra of $\mathcal{K}_N$ correspond to the eigenvalues of the Koopman operator with the associated eigenfunctions converging weakly to an eigenfunction of the Koopman operator, provided that the weak limit of eigenfunctions is nonzero. As a by-product, we propose an analytic version of the EDMD algorithm which, under some assumptions, allows one to construct $\mathcal{K}_N$ directly, without the use of sampling. Finally, under additional assumptions, we analyze convergence of $\mathcal{K}_{N,N}$ (i.e., $M=N$), proving convergence, along a subsequence, to weak eigenfunctions (or eigendistributions) related to the eigenmeasures of the Perron-Frobenius operator. No assumptions on the observables belonging to a finite-dimensional invariant subspace of the Koopman operator are required throughout.