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

2026-08-09

AI Summary

This work addresses the challenge that conventional time-invariant Koopman operators struggle to effectively model frequency-dependent dynamics in non-stationary time series. To overcome this limitation, the authors propose NDKoop, an end-to-end neural architecture that, for the first time, unifies learnable signal decomposition and Koopman-based dynamic modeling within a single framework. By explicitly separating the trend component (frequency-independent) from periodic components (frequency-dependent) and modeling each with dedicated Koopman networks, NDKoop achieves a more accurate representation of non-stationary dynamics. Extensive experiments demonstrate that NDKoop significantly outperforms existing methods across multiple time series forecasting benchmarks, confirming the efficacy and superiority of the proposed decomposition strategy in scenarios where ideal linearization assumptions do not hold.

Abstract

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.

Problem

• Koopman operator
• time-series forecasting
• non-stationary signals
• frequency-dependent dynamics
• signal decomposition

Innovation

• Koopman operator
• neural decomposition
• time-series forecasting
• frequency-dependent dynamics
• end-to-end learning