The Projected Koopman Operator Approximation framework for constructing Filtered EDMD operators is introduced, and under independent noiseless sampling and exact-rank identifiability, the resulting empirical operators converge almost surely to their population counterparts.
Abstract
The Koopman operator provides a linear framework for analyzing nonlinear dynamical systems through spectral properties. Extended Dynamic Mode Decomposition (EDMD) approximates this operator from data, but non-invariant dictionaries can introduce spurious eigenvalues. We introduce the Projected Koopman Operator Approximation framework for constructing Filtered EDMD operators. The framework projects the Koopman action onto admissible dictionary subspaces that need not be invariant, while exactly preserving every represented Koopman eigenpair with nonzero eigenvalue. A forward--intersection chain provides a canonical hierarchy of compatible subspaces, connecting the full dictionary to its maximal invariant core while retaining useful intermediate models. We analyze two projection geometries: a coordinate-orthogonal projector, which requires no function-space Gram-matrix estimate but is basis-dependent, and a function-space orthogonal projector, which recovers population EDMD at the unfiltered level. We characterize their relationship to EDMD and existing subspace-selection methods. We also develop SVD-based algorithms for constructing sampled forward--intersection spaces and implementing the coordinate projector. Under independent noiseless sampling and exact-rank identifiability, the resulting empirical operators converge almost surely to their population counterparts. Experiments on a Kronecker flow, a polynomial system, and the Van~der~Pol oscillator demonstrate reduced spectral pollution. For Van~der~Pol, the coordinate projector recovers the local equilibrium spectrum independently of the sampling measure, whereas the $L^2(\mu)$ projector approximates the limit-cycle spectrum on the same certified subspace.
Fuzzy Spectral Region Decomposition (fSRD), a fully automated learning framework for estimating finite Koopman representation via multiple operators, and a data-adaptive framework for assembling locally invariant embeddings, termed Invariant Decomposition are introduced.
C. Bokor, M. Cary, Denise Morrey et al.· 0 citations
A novel data-driven algorithm to approximate the dominant eigenfunctions of the Koopman operator of nonlinear dynamical systems using neural networks using neural networks to fight the curse of dimensionality arising from using expressive templates for the mode approximation.
Guillaume O. Berger, Raphael M. Jungers· 0 citations
Koopman operator theory enables global linear representations of nonlinear dynamic systems and has gained substantial prominence in orbital dynamics. This paper focuses on a fundamental nonlinear problem in orbital dynamics: the initial value problem of motion around a central celestial body. In practice, an approximate invariant subspace of the Koopman operator should be constructed using a dictionary to realize its finite-dimensional approximation. However, traditional dictionaries are prone to the curse of dimensionality when high precision is required. To address this issue, a novel dictionary based on orbital elements is proposed for the initial value problem. By exploiting the intrinsic dynamic structure of orbital elements, the proposed dictionary effectively mitigates the curse of dimensionality while ensuring high-precision continuous orbital propagation. Numerical simulations demonstrate the feasibility and superiority of the proposed dictionary. On this basis, a new Koopman-operator-based framework is developed for computing and analyzing invariant manifolds of large-scale periodic orbits around asteroids. More significantly, the stickiness effect near invariant manifolds is numerically validated for the first time under a high-precision asteroid gravitational field model. This work provides an efficient dictionary for Koopman operator approximation in orbital dynamics and establishes a critical foundation for invariant manifold studies and low-energy trajectory design in future space missions.
Zihan Liu, Fanghua Jiang· Journal of Guidance Control...· 0 citations
We introduce an intrinsic spectral sparsity model for nonparametric density estimation on compact connected Riemannian manifolds. Instead of penalizing coefficients in an arbitrarily chosen Laplace--Beltrami eigenbasis, we group each complete eigenspace and measure the Hilbert norm of its spectral component. The resulting block-variation space is basis independent and isometry invariant. We establish its structural, atomic, and nonlinear approximation properties and clarify its relation to Sobolev, Besov, and coefficientwise spectral $\ell^1$ classes. We then construct a coordinate-free block-shrinkage estimator and prove a nonasymptotic signal-dependent $L^2$-oracle inequality that adapts to the unknown set of detectable eigenspaces. Under polynomial spectral growth, the risk theory separates the number of spectral blocks from their multiplicities and exhibits two regimes: one driven by a single high-dimensional eigenspace and the other by cumulative spectral complexity. Under matching spectral-growth and nondegeneracy assumptions, corresponding minimax lower bounds show that this multiplicity dependence is intrinsic, with sharp consequences for spheres and the rotation group $SO(3)$. Finally, we develop a positive, normalized, block-penalized exponential spectral sieve for log-densities and derive likelihood oracle inequalities together with expected Kullback--Leibler, Hellinger, and $L^2$ risk bounds. The resulting framework provides a geometry-respecting theory of sparse density estimation that remains invariant under changes of eigenbasis.
Kernel methods, which embed data distributions into a reproducing kernel hilbert space (RKHS) via positive-definite similarity measures, continue to play an important role. However, learning a good, generalizable kernel for high-dimensional and heterogeneous data under temporal or regional distribution shift remains challenging. To address these issues, we propose SpectraMancer, which learns kernels directly in the Fourier spectral domain induced by multilevel circulant matrices, thereby enabling generalizable kernel learning for complex data. SpectraMancer embeds all shift-invariant candidates into a common multilevel order via randomized multilevel circulant matrices, which yields a fixed Fourier diagonalization and turns inverses, products, and gradients into elementwise fast Fourier transform (FFT) operations. To the best of our knowledge, this is the first kernel-learning approach that exploits randomized multilevel circulant matrices for joint diagonalization across kernels. SpectraMancer further enforces scale invariance via kernel double centering and Frobenius normalization, reduces spectral variance through antithetic phase pairing with quasi-Monte Carlo draws, and optimizes a solver-free spectral risk proxy (SRP) for bandwidth weighting without repeated inner solves. Experimental results show that SpectraMancer improves spectrum-aware kernel selection and predictive performance across diverse benchmarks.
Lizhong Ding, Jiarun Fu, Qiuning Wei et al.· IEEE Transactions on Neural...· 0 citations