Orthogonal Koopman Modes

• Orthogonal Koopman modes are Koopman-based modal objects that are mutually orthogonal under specifically chosen inner products or spectral pairings, enabling clear modal decompositions.
• They are computed using methods such as Hankel-DMD, Koopman-LTI, and Schur decompositions, which enhance numerical stability and reduce model order.
• Applications in fluid dynamics and nonlinear PDEs illustrate their power in precise energy attribution and efficient reduced-order modeling.

Searching arXiv for recent and foundational papers on orthogonal Koopman modes, including Koopman–resolvent, Koopman-LTI, Koopman-Schur, Rigged DMD, and randomized orthogonal decompositions. Orthogonal Koopman modes are Koopman-based modal objects that are mutually orthogonal under a specified inner product or generalized spectral pairing. In the standard Koopman formulation, if $u(t)=\Phi^t(u_0)$ solves $\partial_t u=f(u)$, then the operator $K^t$ acts on observables by composition, $(K^t h)(u)=h(\Phi^t(u))$, and an observable can be expanded as $g(u(t))=\sum_n \phi_n(u_0)\,g_n\,e^{i\omega_n t}$, where $g_n$ are Koopman modes. The adjective “orthogonal” enters when those modes, or the associated eigenfunctions, are constructed so that they satisfy orthogonality relations in $L^2$, in space–time inner products, in reduced data spaces, or in rigged Hilbert spaces. The resulting objects appear in several distinct but related settings: translation-invariant travelling waves, QR/SVD-based Koopman-LTI constructions, Galerkin discretizations on orthogonal polynomial bases, Schur-based invariant-subspace decompositions, resolvent-mode analysis of the Navier–Stokes equations, and generalized eigenfunction decompositions for continuous spectra (Sharma et al., 2016, Li et al., 2021, Drmač et al., 2023, Colbrook et al., 2024).

1. Koopman-theoretic setting

The Koopman operator is a linear operator on observables, not on the state variables themselves. In discrete time, for a measure-preserving dynamical system $(M,T,\mu)$, it acts on $L^2(M,\mu)$ by $(Uf)(x)=f(T(x))$. In continuous time, the corresponding flow map $\partial_t u=f(u)$0 yields $\partial_t u=f(u)$1. In either case, an eigenfunction $\partial_t u=f(u)$2 satisfies $\partial_t u=f(u)$3 or $\partial_t u=f(u)$4, and the evolution of an observable can be represented spectrally in terms of eigenvalues, eigenfunctions, and associated physical-coordinate modes (Arbabi et al., 2016, Sharma et al., 2016).

For spatially extended systems, the Koopman framework can be enlarged to include symmetry actions. A spatial shift operator is defined by

$\partial_t u=f(u)$5

and the combined spatio-temporal Koopman operator by

$\partial_t u=f(u)$6

Its eigenfunctions satisfy

$\partial_t u=f(u)$7

so that an invariant field admits the travelling-wave expansion

$\partial_t u=f(u)$8

This formulation makes explicit that Koopman analysis can incorporate not only time translation but combinations of symmetry operations and temporal evolution (Sharma et al., 2016).

A parallel line of work treats the Koopman operator through finite-dimensional compressions derived from data. In Hankel-DMD and EDMD-type settings, one approximates the action of the operator on a finite dictionary or Krylov subspace. Under ergodicity, measure preservation, or the existence of a physical measure, the data-driven projections converge to Hilbert-space projections, and the computed DMD eigenpairs converge to Koopman eigenpairs on invariant subspaces (Arbabi et al., 2016).

2. Sources of orthogonality

Orthogonality in Koopman analysis is not a single theorem but a family of constructions tied to the chosen function space, inner product, and representation. In the spatio-temporal setting of translation-invariant flows, the eigenfunctions $\partial_t u=f(u)$9 are orthogonal under the space-time inner product, so the travelling-wave exponentials $K^t$0 form an orthonormal basis. In those invariant directions, Koopman modes reduce to mutually orthogonal Fourier modes (Sharma et al., 2016).

A different mechanism appears in the Koopman-LTI architecture for fluid–structure systems. There the retained invariant subspace is endowed with an inner product

$K^t$1

and right and left spectral objects satisfy the bi-orthogonality relation

$K^t$2

The construction explicitly normalizes the retained modes so that the Koopman modes themselves become orthogonal in the fluid–structure domain (Li et al., 2021).

In ergodic $K^t$3 settings, orthogonality may arise from an orthonormal eigenbasis of the Koopman operator. If $K^t$4 is an orthonormal set of eigenfunctions satisfying $K^t$5, then any $K^t$6 admits the Parseval-type expansion

$K^t$7

Arbabi and Mezić show that, with Hankel data and infinite-time observations, DMD converges to these Koopman eigenfunctions and eigenvalues on finite-dimensional invariant subspaces, while the singular vectors converge to POD directions in $K^t$8 (Arbabi et al., 2016).

Orthogonality can also be generalized beyond pure point spectrum. Rigged DMD places the Koopman operator in a Gelfand triple $K^t$9 and constructs generalized eigenfunctions $(K^t h)(u)=h(\Phi^t(u))$0. After renormalization, the resulting family satisfies

$(K^t h)(u)=h(\Phi^t(u))$1

providing an orthogonality relation appropriate to continuous spectral components (Colbrook et al., 2024).

This body of work suggests that orthogonal Koopman modes are best understood as representation-dependent objects: orthogonality may hold for exact eigenfunctions, for physical-coordinate modes, for Schur vectors spanning invariant subspaces, or for generalized eigenfunctions, depending on the analytic setting.

3. Computational constructions

Several computational frameworks enforce or recover orthogonality in Koopman analysis.

Framework	Core computation	Orthogonality mechanism
Hankel-DMD (Arbabi et al., 2016)	Reduced SVD of Hankel data and eigendecomposition of $(K^t h)(u)=h(\Phi^t(u))$2	Convergence of SVD to POD basis in $(K^t h)(u)=h(\Phi^t(u))$3
Koopman-LTI (Li et al., 2021)	Global SVD, low-dimensional propagator, then thin-QR or SVD of modes $(K^t h)(u)=h(\Phi^t(u))$4	$(K^t h)(u)=h(\Phi^t(u))$5 gives exactly orthonormal modes
Symmetry-restricted DMD (Sharma et al., 2016)	Apply group action $(K^t h)(u)=h(\Phi^t(u))$6 in $(K^t h)(u)=h(\Phi^t(u))$7	Modes lie in invariant subspace of symmetry group
Legendre Galerkin (Servadio et al., 2021)	Generalized eigenproblem in orthonormal polynomial basis	$(K^t h)(u)=h(\Phi^t(u))$8 under orthonormal Legendre basis
Koopman-Schur EDMD (Drmač et al., 2023)	EDMD compression followed by real or complex Schur form	Unitary or orthogonal Schur vectors
KROD (Bistrian, 5 Aug 2025)	Randomized SVD, Koopman compression, eigendecomposition of $(K^t h)(u)=h(\Phi^t(u))$9	$g(u(t))=\sum_n \phi_n(u_0)\,g_n\,e^{i\omega_n t}$0 with $g(u(t))=\sum_n \phi_n(u_0)\,g_n\,e^{i\omega_n t}$1
Rigged DMD (Colbrook et al., 2024)	mpEDMD plus resolvent-based wave-packet filtering	Orthonormal generalized eigenfunctions in rigged space

In the ergodic Hankel-DMD approach, one builds a Hankel matrix from a time series, computes $g(u(t))=\sum_n \phi_n(u_0)\,g_n\,e^{i\omega_n t}$2, forms $g(u(t))=\sum_n \phi_n(u_0)\,g_n\,e^{i\omega_n t}$3, and solves $g(u(t))=\sum_n \phi_n(u_0)\,g_n\,e^{i\omega_n t}$4. The DMD modes are $g(u(t))=\sum_n \phi_n(u_0)\,g_n\,e^{i\omega_n t}$5, and Birkhoff’s ergodic theorem justifies replacing function-space inner products by long-time vector inner products along a trajectory. The paper further shows that the SVD central to DMD converges to the POD of observables, furnishing an orthonormal basis in $g(u(t))=\sum_n \phi_n(u_0)\,g_n\,e^{i\omega_n t}$6 when the required invariant-subspace assumptions hold (Arbabi et al., 2016).

In Koopman-LTI, orthogonality is enforced directly after mode reconstruction. Starting from mean-subtracted snapshot matrices $g(u(t))=\sum_n \phi_n(u_0)\,g_n\,e^{i\omega_n t}$7 and $g(u(t))=\sum_n \phi_n(u_0)\,g_n\,e^{i\omega_n t}$8, one computes the reduced SVD $g(u(t))=\sum_n \phi_n(u_0)\,g_n\,e^{i\omega_n t}$9, forms $g_n$0, solves $g_n$1, reconstructs full modes $g_n$2, and finally applies a thin-QR or SVD,

$g_n$3

to obtain orthonormal modes $g_n$4 with $g_n$5. The reported consequence is a basis of Koopman modes that is exactly orthonormal within machine precision and sampling-independent once $g_n$6 spans one fundamental period (Li et al., 2021).

A Galerkin alternative uses an orthogonal Legendre basis. For a polynomial ODE, one expands each eigenfunction as $g_n$7, imposes the Koopman eigenfunction PDE through Galerkin projection, and obtains

$g_n$8

because the mass matrix is the identity in an orthonormal basis. The resulting Koopman modes for the state observable are $g_n$9. The tutorial emphasizes that orthogonality of the basis removes ill-conditioning from Gram-matrix inversion and improves interpretability by reducing modal leakage (Servadio et al., 2021).

Two newer approaches shift the emphasis from eigenvectors to orthogonal subspaces. The Koopman-Schur framework replaces a potentially ill-conditioned diagonalization $L^2$0 by a Schur factorization $L^2$1, or by a real Schur form $L^2$2 for real data. The first columns of $L^2$3 span invariant subspaces associated with selected eigenvalues, and the modal analysis is carried out in this orthonormal basis of Schur vectors rather than in possibly unstable eigenvectors (Drmač et al., 2023). The randomized Koopman Orthogonal Decomposition constructs $L^2$4 via randomized SVD, forms $L^2$5, diagonalizes the Hermitian Gram matrix $L^2$6, and sets

$L^2$7

This yields orthonormal spatial Koopman modes together with projected amplitudes $L^2$8 (Bistrian, 5 Aug 2025).

4. Resolvent operators, symmetries, and orthogonal mode optimality

A central fluid-mechanical formulation connects Koopman mode decomposition to the resolvent operator of the linearized incompressible Navier–Stokes equations. About a time-mean flow $L^2$9, the fluctuating velocity $(M,T,\mu)$0 satisfies

$(M,T,\mu)$1

where $(M,T,\mu)$2 is the fluctuating Reynolds-stress divergence. After temporal Fourier transform and Leray projection, one obtains

$(M,T,\mu)$3

with

$(M,T,\mu)$4

Equivalently,

$(M,T,\mu)$5

The singular value decomposition

$(M,T,\mu)$6

produces orthonormal forcing modes $(M,T,\mu)$7, orthonormal response modes $(M,T,\mu)$8, and gains $(M,T,\mu)$9 satisfying $L^2(M,\mu)$0 and $L^2(M,\mu)$1 (Sharma et al., 2016).

The significance of this SVD is twofold. First, it is the Schmidt decomposition of the resolvent and is optimal in the operator or Frobenius norm: $L^2(M,\mu)$2 Second, when the true velocity-field Koopman mode $L^2(M,\mu)$3 is unknown because the corresponding forcing is unknown, the leading response modes provide the best $L^2(M,\mu)$4-term orthonormal approximation,

$L^2(M,\mu)$5

Accordingly, the columns of $L^2(M,\mu)$6 form an optimal orthonormal basis for representing unknown velocity-field Koopman modes (Sharma et al., 2016).

The same paper generalizes DMD by incorporating symmetry actions into the snapshot relation. Instead of $L^2(M,\mu)$7, one uses

$L^2(M,\mu)$8

where $L^2(M,\mu)$9 may be a spatial shift, a reflection, or a combined travelling-wave action $(Uf)(x)=f(T(x))$0. The resulting DMD modes are restricted to the invariant subspace of the symmetry group and inherit the associated orthogonality properties, notably Fourier orthogonality in shift-invariant directions (Sharma et al., 2016).

The practical procedure reported for this orthogonal approximation consists of computing the time-mean flow, choosing frequencies $(Uf)(x)=f(T(x))$1, forming $(Uf)(x)=f(T(x))$2, performing its SVD, truncating to leading response modes, and then using those modes as an orthonormal basis for approximating unknown Koopman modes. If data are available, one may additionally enforce spatio-temporal symmetries by generalized DMD and reconstruct the full field through inverse Fourier or spectral sums (Sharma et al., 2016).

5. Representative applications

In fluid–structure interaction, orthogonal Koopman modes have been used to identify deterministic excitation–response mechanisms. The Koopman-LTI analysis of a subcritical prism wake reports six corresponding, orthogonal, and in-synch fluid excitation–structure response mechanisms. The retained modes are $(Uf)(x)=f(T(x))$3 at $(Uf)(x)=f(T(x))$4, $(Uf)(x)=f(T(x))$5 at $(Uf)(x)=f(T(x))$6, $(Uf)(x)=f(T(x))$7 at $(Uf)(x)=f(T(x))$8, $(Uf)(x)=f(T(x))$9 at $\partial_t u=f(u)$00, $\partial_t u=f(u)$01 at $\partial_t u=f(u)$02, and $\partial_t u=f(u)$03 at $\partial_t u=f(u)$04. They are interpreted, respectively, as primary shear-layer or shedding dynamics, low-frequency turbulence production, a subharmonic, a second harmonic, an ultra-harmonic at $\partial_t u=f(u)$05, and a 2P-type embedded ultra-harmonic. Because the orthogonality relation $\partial_t u=f(u)$06 yields decoupled amplitudes $\partial_t u=f(u)$07, the total energy becomes $\partial_t u=f(u)$08, enabling mode-by-mode attribution of phenomena such as downstream-wall lift peaks and low-frequency drag fluctuations. The same framework identifies “vortex breathing” as a modal energy exchange in which one mode grows while another decays and the total modal energy remains constant (Li et al., 2021).

Ergodic-theoretic DMD provides a complementary set of applications. Reported demonstrations include periodic flow in a lid-driven cavity at $\partial_t u=f(u)$09, where DMD finds frequencies $\partial_t u=f(u)$10 with $\partial_t u=f(u)$11; quasi-periodic flow at $\partial_t u=f(u)$12, where Hankel-DMD locates a 2-torus spectrum; the Lorenz attractor, for which the Hankel SVD yields an orthonormal basis of observables even though only the trivial eigenvalue is discrete; and the Van der Pol oscillator, where a two-trajectory construction recovers the basic frequency and an eigenfunction whose level sets are asymptotic isochrons (Arbabi et al., 2016).

Orthogonal Koopman modes have also been used for nonlinear PDE reduction. The KROD framework is demonstrated on three viscous Burgers’ shock-wave experiments on $\partial_t u=f(u)$13 with $\partial_t u=f(u)$14, $\partial_t u=f(u)$15, and $\partial_t u=f(u)$16. The experiments use the initial data $\partial_t u=f(u)$17, a Riemann problem with step initial condition $\partial_t u=f(u)$18, and $\partial_t u=f(u)$19. Pareto-front selection gives $\partial_t u=f(u)$20, $\partial_t u=f(u)$21, and $\partial_t u=f(u)$22 for the three cases. The paper reports $\partial_t u=f(u)$23 with $\partial_t u=f(u)$24, $\partial_t u=f(u)$25, and $\partial_t u=f(u)$26, respectively, and the Modal Assurance Criterion matrix is stated to be the identity up to numerical precision, with zeros off diagonal and ones on diagonal. The orthogonal spatial modes are coupled to an NLARX model for the temporal coefficients, and the reported validation uses only one-third of the snapshots for training while maintaining accurate reproduction of training and unseen data (Bistrian, 5 Aug 2025).

6. Interpretation, numerical advantages, and conceptual boundaries

The principal numerical advantages claimed for orthogonal Koopman modes are stability, compactness, and interpretability. In the resolvent setting, orthonormal response modes provide the best rank-$\partial_t u=f(u)$27 approximation to unknown velocity-field Koopman modes when the nonlinear forcing is not known a priori (Sharma et al., 2016). In Koopman-LTI, a single global SVD followed by a single QR yields a basis that is exactly orthonormal within machine precision and sampling-independent once the sampling interval resolves the fundamental spectral gaps (Li et al., 2021). In the randomized orthogonal decomposition, orthogonality guarantees minimal redundancy and optimal representation in the $\partial_t u=f(u)$28 sense, while the paper further states that the method can deliver one- to two-orders-of-magnitude model-order reduction compared to classical DMD and is fully self-consistent, with Pareto-front analysis used for automatic rank selection (Bistrian, 5 Aug 2025).

A parallel numerical motivation is conditioning. The Koopman-Schur framework starts from the observation that DMD and EDMD may produce highly ill-conditioned eigenvector matrices when the finite-dimensional Koopman compression is nonnormal or nearly defective. By replacing eigenvector-based modes with invariant-subspace modes derived from unitary or orthogonal Schur transformations, the method avoids unstable inversions and permits ordered selection of spectral components through partial ordered Schur decompositions (Drmač et al., 2023). The Legendre-Galerkin formulation reaches a related goal from the opposite direction: by choosing an orthonormal polynomial basis, the mass matrix is the identity, Gram-matrix inversion is avoided, and the eigenproblem becomes better conditioned (Servadio et al., 2021).

The literature also imposes clear conceptual boundaries. Orthogonality does not imply that all constructions are describing the same object. In some works the orthogonal entities are Koopman eigenfunctions in $\partial_t u=f(u)$29; in others they are physical-coordinate modes, resolvent response modes, Schur vectors spanning invariant flags, or generalized eigenfunctions indexed continuously by frequency. This suggests that “orthogonal Koopman modes” is a family resemblance term rather than a uniquely fixed one. The distinction is especially important for systems with continuous spectrum: classical eigenfunction-based DMD can be inadequate there, whereas Rigged DMD replaces point-spectrum modes by wave-packet approximations to generalized eigenfunctions and proves high-order convergence results for the corresponding spectral measures and generalized modes (Colbrook et al., 2024).

A related misconception is that orthogonality is automatic once one invokes Koopman theory. The cited works instead indicate that orthogonality must typically be induced by symmetry, by an explicitly chosen inner product, by bi-orthogonal left–right constructions, by SVD or QR post-processing, by Schur-unitary factorizations, or by rigged Hilbert-space machinery. The common theme is not automatic diagonalization of nonlinear dynamics, but the disciplined construction of modal coordinates in which decomposition, projection, and reconstruction become technically tractable and physically interpretable across discrete, continuous, and symmetry-reduced spectral settings (Sharma et al., 2016, Li et al., 2021, Drmač et al., 2023, Colbrook et al., 2024).