Learnable Koopman-Enhanced Transformer-Based Time Series Forecasting with Spectral Control

TL;DR

This paper introduces learnable Koopman operator parameterizations integrated with transformer-based models for time series forecasting, enabling spectral control, stability, and interpretability, and demonstrating improved performance and theoretical grounding.

Contribution

It presents four novel learnable Koopman variants that unify linear dynamical systems with deep learning, allowing explicit spectral, stability, and rank control within forecasting architectures.

Findings

Enhanced forecasting accuracy across multiple benchmarks.

Improved spectral stability and interpretability of latent dynamics.

Demonstrated effectiveness of Koopman operators in deep models.

Abstract

This paper proposes a unified family of learnable Koopman operator parameterizations that integrate linear dynamical systems theory with modern deep learning forecasting architectures. We introduce four learnable Koopman variants-scalar-gated, per-mode gated, MLP-shaped spectral mapping, and low-rank Koopman operators which generalize and interpolate between strictly stable Koopman operators and unconstrained linear latent dynamics. Our formulation enables explicit control over the spectrum, stability, and rank of the linear transition operator while retaining compatibility with expressive nonlinear backbones such as Patchtst, Autoformer, and Informer. We evaluate the proposed operators in a large-scale benchmark that also includes LSTM, DLinear, and simple diagonal State-Space Models (SSMs), as well as lightweight transformer variants. Experiments across multiple horizons and patch lengths show that learnable Koopman models provide a favorable bias-variance trade-off, improved conditioning, and more interpretable latent dynamics. We provide a full spectral analysis, including eigenvalue trajectories, stability envelopes, and learned spectral distributions. Our results demonstrate that learnable Koopman operators are effective, stable, and theoretically principled components for deep forecasting.