Extended dynamic mode decomposition

Core idea

Choose observables $\psi_1,\ldots,\psi_r$ and sample pairs $y_m=T(x_m)$ . Extended dynamic mode decomposition (EDMD) approximates each $U_T\psi_j$ by $\sum_iK_{ij}\psi_i$ through empirical least squares. Define

(G_M)_{ij}=\frac1M\sum_m\overline{\psi_i(x_m)}\psi_j(x_m),\qquad (C_M)_{ij}=\frac1M\sum_m\overline{\psi_i(x_m)}\psi_j(y_m).

If $G_M$ is invertible, the coefficient matrix is $K_M=G_M^{-1}C_M$ . A singular Gram matrix requires a specified least-squares solution or regularization.

Population operator

For linearly independent $L^2(\mu)$ observables, replace the empirical sums by integrals. The resulting matrix represents $P_VU_T|_V$ , with $V=\operatorname{span}\{\psi_j\}$ and $P_V$ the orthogonal projection. It is exact on $V$ when $U_TV\subseteq V$ .

Ergodic sampling

If $T$ preserves an ergodic probability and $x_m=T^mx_0$ , Birkhoff's theorem makes these finitely many Gram and cross-moment entries converge for almost every $x_0$ . Their integrability follows from Cauchy–Schwarz. If the population Gram matrix is invertible, the empirical matrices are eventually invertible and $K_M$ converges to the fixed-dictionary population matrix.

This is not a general theorem of convergence of all fitted eigenvalues to the spectrum of the infinite-dimensional Koopman operator as the dictionary grows.

References

M. O. Williams, I. G. Kevrekidis, and C. W. Rowley, “A Data-Driven Approximation of the Koopman Operator: Extending Dynamic Mode Decomposition”, §§2–3; the trajectory-sampling statement follows here by Birkhoff.