Deep Learning Enhanced Dynamic Mode Decomposition

The paper’s Theorem 2 asserts that, under Ansatz 1 (dictionary invariance) and diagonalizable Ċ(0), EDMD “only computes spectra and eigenfunctions of K_t,” deriving a finite-dimensional connection-matrix dynamics Ċ(t)=Ċ(0)C(t) and concluding C(t)=Ṽ e^{tΛ} Ṽ^{-1} (Theorem 2 and its proof) . It also states the standard EDMD pipeline (least-squares problem Ko=argmin ||Ψ+−KΨ−||^2_F, SVD-based formula, spectral readout λ=ln t̃/δt, and eigenfunction reconstruction) . However, the paper stops short of a precise identification Ko=C(δt) and does not state the usual data richness/full row-rank condition on Ψ− that guarantees EDMD recovers the exact finite-dimensional Koopman matrix; instead it argues informally that, since r(x;K)=0 under Ansatz 1, the constructions “must be equivalent” to the EDMD minimizer . The candidate solution supplies the missing technical step: with noiseless on-trajectory snapshots, Ψ+=C(δt)Ψ−, hence Ko=Ψ+(Ψ−)+=C(δt)Π and, when rank(Ψ−)=N, Ko=C(δt). It then explicitly identifies eigenpairs, clarifies the left-eigenvector/eigenfunction connection, and addresses branch selection for the logarithm. Thus the model’s solution is correct and more complete; the paper’s result is essentially correct but omits the standard rank/coverage hypothesis needed to make the EDMD⇔C(δt) equivalence rigorous.