Koopman theory offers a linear-operator view of nonlinear sequence dynamics by lifting observations into a space where evolution is governed by a linear time-invariant Koopman operator. While the Koopman operator provides a linear representation of nonlinear dynamics, it is generally infinite dimensional and defined under time-invariant assumptions. To model non-stationary signals with frequency-dependent behavior, a frequency-varying extension is required. In recent years, deep learning has been increasingly employed to exploit its powerful function-approximation ability for learning the Koopman operator. In this study, we propose a novel approach called neural decomposition Koopman (NDKoop), an end-to-end architecture that integrates a learnable signal decomposition module with both frequency-independent and frequency-dependent Koopman based networks for sequence forecasting. To the best of our knowledge, this is the first work to jointly realize end-to end Koopman modeling and signal decomposition within a unified neural framework. We demonstrate that decomposing a signal into a frequency-independent trend component and a frequency-dependent periodic component, each governed by a corresponding Koopman operator, improves prediction accuracy when perfect linearization is unattainable. Numerical experiments across several forecasting benchmarks indicate that the proposed NDKoop provides strong performance.
Wrist-worn photoplethysmography (PPG) enables continuous monitoring of cardiopulmonary physiology, but reliable heart rate (HR) and respiratory rate (RR) estimation in free-living conditions remains challenging due to non-stationary motion artifacts that spectrally overlap with physiological dynamics. Existing signal-processing methods degrade under strong motion, while unconstrained deep learning approaches often lack physiological interpretability and identifiable structure. We propose a Physically-Constrained Harmonic Separation (PCHS) framework that formulates HR and RR estimation from wrist PPG as an analysis-by-synthesis problem, where accelerometer measurements condition artifact separation rather than directly regressing vital signs. A physics-guided harmonic generator decomposes the observed signal into quasi-periodic physiological components and a motion-related residual, enabling HR recovery from the fundamental frequency and RR prediction from respiratory-driven modulations of the harmonic parameters. Robust reconstruction objectives, separation constraints, and uncertainty-aware weighting stabilize the decomposition under motion. Experiments on the motion-intensive PPG-DaLiA dataset demonstrate that PCHS outperforms state-of-the-art methods while yielding interpretable signal decompositions that effectively disentangle physiological activity from motion artifacts.
Interval wind speed forecasting is essential for the efficient integration of wind energy into power systems, as it accounts for the inherent uncertainty of wind resources. This study presents a systematic literature review focused on hybrid approaches to interval forecasting of wind generation, exploring the combination of deep learning, modal decomposition, and statistical methods. To guide the paper selection, Latent Dirichlet Allocation (LDA) was applied for topic modeling, enabling the identification of patterns and research trends. The findings emphasize that integrating hybrid models with decomposition techniques-such as Variational Mode Decomposition (VMD) and Ensemble Empirical Mode Decomposition (EEMD)-enhances forecast accuracy and reliability by narrowing prediction intervals without compromising coverage. Regarding interval construction, most studies adopt a dual-model strategy, independently forecasting the lower and upper bounds. Input data are commonly decomposed using techniques like EMD, EEMD, or VMD, which extract frequency-based components. These components serve as inputs to models such as LSTM or ELM, trained separately for each bound. This approach allows for targeted modeling of uncertainty, improving flexibility and precision, Interval quality is typically evaluated through metrics that balance coverage and interval width. The review also highlights challenges, including the lack of standardized evaluation metrics, computational complexity, and limited real-world validation. Overall, the study reinforces the value of interval forecasting for wind energy operations and offers insights for advancing model robustness and decision-making.