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.
Abstract
Highly nonlinear chaotic dynamical systems remain difficult to model due to fundamental trade-offs between complexity, expressivity, and data efficiency. Modern machine learning methods achieve strong predictive performance but often rely on a-priori system knowledge or curated data with limited interpretability. Koopman operator theory offers a promising direction via linear representation in an infinite-dimensional observable space. However, many data-driven Koopman methods seek globally valid operators for which useful finite-dimensional spectral embeddings remain difficult to identify under these constraints. To overcome associated limitations, we introduce Fuzzy Spectral Region Decomposition (fSRD), a fully automated learning framework for estimating finite Koopman representation via multiple operators. The proposed method realizes a data-adaptive framework for assembling locally invariant embeddings, termed Invariant Decomposition. fSRD achieves highly accurate linear reconstructions of nonlinear systems while learning finite-dimensional representations of their induced evolution operators, bridging interpretable operator-theoretic models with expressive data-driven sequence learning. These embeddings are adaptively constructed via a global fuzzy tree model, drawing inspiration from fuzzy neural architectures to learn the induced dynamics while prioritizing parsimonious solutions. Empirical results across canonical chaotic systems (e.g., Lorenz and Duffing) and high-dimensional real-world data demonstrate strong predictive accuracy, interpretability, and robust expressivity across data-rich and data-limited regimes, highlighting the method's generality.
One of the main objectives in control theory is to obtain a linear representation of inherently nonlinear systems in order to leverage the analytical and theoretical tools developed for linear systems. In this context, the Koopman operator has attracted increasing interest in recent years.Koopman operator theory provides a framework in which nonlinear dynamical systems are represented by a linear operator acting on an infinite-dimensional Hilbert space. Since such an infinite-dimensional representation is not numerically tractable, numerous finite-dimensional approximation methods have been proposed. These approaches typically rely on time-series data and include extended dynamic mode decomposition as well as deep learning–based variants. In this paper, we propose an original machine-learning-based approach for the synthesis of a fixed-dimensional Koopman approximant (lifting) of continuous-time nonlinear systems. A differential state-space representation of the system (as opposed to a recurrent state model) is assumed to be available through its vector field (f). The proposed encoder departs from conventional approaches in that it does not directly output the current latent state, but instead generates samples of the latent trajectory evaluated at user-defined time instants (temporal discretization). This formulation enables the integration into the learning process of Physical & Latent Continuous Losses, enforcing consistency between the physical dynamics and the Koopman dynamics, as well as Physical & Latent Boundary Losses, ensuring consistency with the prescribed initial conditions. In parallel, we introduce a structural stability constraint on the Koopman operator. The effectiveness of the proposed methodology is demonstrated through the analysis and simulation of two polynomial dynamical systems.
M. Zodros, A. Colotti, M. Yagoubi et al.· International Conference on...· 0 citations
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.
We develop operator-theoretic generalization bounds for deep multi-output function classes by representing network layers as Koopman composition operators on vector-valued reproducing kernel Hilbert spaces. In vector-valued Sobolev RKHSs, we derive Rademacher complexity bounds for invertible and width-expanding injective architectures. The estimates separate the output-coupling contribution, represented by the trace of the task matrix, from the layerwise operator norms, Sobolev symbol ratios, determinant factors, and restriction constants generated by the linear maps. We then analyze a distinct one-dimensional Brownian/Cameron--Martin regime. Using the exact anchored derivative-norm characterization of the vector-valued Brownian RKHS, we obtain layerwise bounds for domain-preserving scalar linear maps and anchored diffeomorphic activations; the corresponding factors scale as $|W_l|^{1/2}$ and $\|\sigma_l'\|_\infty^{1/2}$, respectively, and do not involve Sobolev smoothness exponents. Because the Sobolev and Brownian results concern different hypothesis spaces, neither is asserted to dominate the other uniformly. We additionally formulate shared operator learning across tasks, prove a finite-rank representer theorem, derive the exact finite-dimensional problem for squared loss, and establish a target-transfer bound when the learned operator is obtained independently of the target sample. Synthetic and MNIST studies examine stabilized Sobolev-inspired and Brownian-inspired complexity proxies; these empirical proxies are not evaluations of the proved bounds for rank-deficient architectures.
Mahdi Mohammadigohari, Thomas Borsani, G. D. Fatta· 0 citations
A physics-informed Koopman representation based on generalized momenta is introduced, yielding a linear control-affine model in lifted coordinates with known input structure that avoids the bilinear state – input coupling inherent in standard Koopman approaches, enabling improved prediction accuracy and tractable controller synthesis.
Autoencoders are commonly used for unsupervised data-driven fault detection in nonlinear dynamical systems. Despite their widespread success and often favorable performance compared with traditional approaches, most applications rely on heuristic reconstruction of measured data using features learned from nominal training data, without explicit insight into the underlying nonlinear dynamics. This lack of interpretability limits the extension of autoencoder-based fault detection methods to higher levels of fault diagnosis, e.g., fault localization and quantification, and confines their use largely to application-oriented studies. To address this limitation, we propose a strategy for detecting parametric faults in nonlinear stochastic mechanical systems. A mathematical representation of the output data is developed using Koopman operator theory, which motivates their embedding on a manifold and its subsequent approximation with a two-stage autoencoder. Fault detection is formulated within a hypothesis-testing framework, in which new data are tested for consistency with a neighborhood of the manifold identified from nominal observations. The proposed method is validated through Monte Carlo simulations of a toy mechanical system with two types of nonlinearity and applied to two well-known real benchmarks, where it provides favorable fault-detection performance compared with standard autoencoders.
Learning mappings between infinite-dimensional objects is a central challenge in scientific machine learning. We introduce a general kernel-based encoder-decoder framework for operator learning that separates observation, representation, learning, and reconstruction. We develop this framework for multi-input, multi-output operator learning, where operators map between products of potentially distinct function spaces. Our approximation theory shows that, although the number of inputs and outputs can increase, the convergence rate is governed by the most challenging constituent approximation problem rather than the overall problem dimension. The framework leads to practical kernel methods with closed-form training and inference, combining mathematical tractability with computational efficiency. We further specialize the approach to multiple operator learning by introducing KernelMO, a family of kernel methods with complementary operator-valued and product-space formulations. Across five families of parametric partial differential equations, the proposed methods achieve competitive or state-of-the-art predictive accuracy while reducing training and inference costs relative to neural operator architectures and deep learning based models, offering an efficient and lightweight alternative.
Adrien Weihs, Chunyang Liao, Jingmin Sun et al.· 0 citations