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, which provides favorable fault-detection performance compared with standard autoencoders.
Abstract
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.
Identifying stochastic dynamical systems from observational data remains a major challenge in applied mathematics and engineering, particularly when complex systems are influenced by random perturbations and incomplete empirical information. This comprehensive review aims to examine state-of-the-art data-driven methods for discovering governing equations, estimating parameters, and predicting the behavior of stochastic dynamical systems. The review systematically analyzes key methodological approaches, including Sparse Identification of Nonlinear Dynamics (SINDy), Dynamic Mode Decomposition (DMD) and its extensions, Koopman operator theory, neural ordinary differential equations, and Bayesian inference. Each approach is evaluated in terms of its theoretical foundations, computational requirements, robustness to noise, and applicability to different classes of stochastic systems. Drawing on numerical experiments and real-world case studies, the findings show that no single method consistently outperforms others across all scenarios. Instead, hybrid approaches that integrate physics-informed constraints with machine learning demonstrate the strongest potential for advancing data-driven system identification. The review concludes that future research should address real-time identification, uncertainty quantification, and the integration of multi-fidelity data sources to improve the reliability and scalability of stochastic system modeling. This work contributes a comprehensive framework for guiding researchers and practitioners in selecting and implementing appropriate identification methods for stochastic dynamical systems.
Rishav Jha, Kameshwar Sahani, S. K. Sahani et al.· African Multidisciplinary Jo...· 0 citations
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
An algorithmic framework, TAkens Reconstruction (TAR), to analyze and reconstruct arbitrary dynamical systems from low-dimensional time series using an integration of Takens'Delay Embedding Theorem, manifold learning techniques, and universal function approximators is developed.
By focusing on algorithmic stability as a means of establishing out-of-sample bounds, we provide a system-theoretic interpretation of generalization in learning-enabled dynamical systems arising in data-driven optimization and feedback control approximation. Given two neighboring datasets, we specifically model sample replacement as an exogenous disturbance acting on a sensitivity system, while the incremental behavior of the data-dependent operator is encoded through an integral quadratic constraint. By relying on dissipativity arguments, we establish a matrix inequality-based certificate and a uniform stability bound that separates the one-sample sensitivity of the learned operator, and an algorithm-dependent dynamical gain. The latter can then be optimized, offering a tractable tool for certifying and comparing generalization capabilities of learning dynamics. We show that our results recover classical ones for gradient descent, apply naturally to momentum-based methods such as heavy-ball and Nesterov acceleration, and extend to data-driven control.
Rolling bearing fault diagnosis under noise and varying operating conditions requires diagnostic evidence that is not only accurate but also physically verifiable. Existing deep diagnostic models often rely on post-hoc saliency or attention visualization, which may not provide stable correspondence with measurable dynamic quantities. This study proposes a physically compiled Kolmogorov–Arnold network(PC-KAN), in which underdamped impulse-response priors are embedded into front-end filtering, spline-based functional representation, and physics-constrained training, so as to generate mechanism-aligned and verifiable diagnostic evidence. Specifically, the model consists of a physically compiled filter layer (PCFL-CBAM), statistical feature aggregation, feature standardization and input normalization, a MultKAN representation module, and a linear classifier head. The PCFL is physically parameterized using the product of an exponentially decaying envelope and a narrowband oscillation, and its B-spline coefficients are initialized via dynamic-topological mapping using training-domain spectral evidence. Kernel energy constraints and bandwidth constraints are adopted to mitigate the degradation of convolutional kernel morphology and frequency band drift. The convolutional block attention module (CBAM) is employed to enhance the responses at critical channels and informative temporal locations. A three-stage freeze-thaw training strategy is applied to sequentially achieve the construction of discriminative mappings and the refinement of physical semantics. Experiments on bearing fault datasets show that the proposed method achieves competitive diagnostic performance while providing interpretable evidence including kernel frequency responses, damping-related parameters, bandwidth concentration, and sparse decision paths. The results indicate that PC-KAN is most valuable as a mechanism-aligned diagnostic framework rather than merely as a highest-accuracy classifier.
Bo Xu, Huipeng Li, Fengxing Zhou et al.· Measurement science and tech...· 0 citations
We study the reconstruction of an unknown dynamical system from a single noisy scalar time series. The goal is to recover the underlying dynamics for forecasting. We introduce a method that uses differential embedding coordinates to identify a rational closure of the embedding dynamics directly from data. The closure is identified through a weak-form regression pipeline, which avoids unstable pointwise differentiation of noisy data. When applied to noise-free Lorenz and R\"ossler systems, the method recovers closures that support long forecasts across a broad ensemble of realizations ($18.1$ and $7.1$ Lyapunov times respectively). Under $15$--$30\%$ additive Gaussian noise, performance becomes system-dependent. For the Lorenz system, forecast horizons remain short even in the best cases, whereas the R\"ossler system generally performs better in absolute terms, though not once normalized by the Lyapunov time. Our proposed method recovers directly interpretable closure coefficients which we compared against the known analytic closures of the Lorenz and R\"ossler systems.