End-to-End Neural Decomposition with Koopman Operators for Time-Series Forecasting

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.

High-Order Synchrosqueezed Chirplet Transforms for Multicomponent Signal Analysis

This study focuses on the analysis of signals containing multiple components with crossover instantaneous frequencies (IF). This problem was initially solved with the chirplet transform (CT). Also, it can be sharpened by adding the synchrosqueezing step, which is called the synchrosqueezed chirplet transform (SCT). However, we found that the SCT goes wrong with the high chirp modulation signal due to the wrong estimation of the IF. In this paper, we present the improvement of the post-transformation of the CT. The main goal of this paper is to amend the estimation introduced in the SCT and carry out the high-order synchrosqueezed chirplet transform. The proposed method reduces the wrong estimation when facing a stronger variety of chirp-modulated multi-component signals. The theoretical analysis of the new reassignment ingredient is provided. Numerical experiments on some synthetic signals are presented to verify the effectiveness of the proposed high-order SCT.