SKOLR: Structured Koopman Operator Linear RNN for Time-Series Forecasting

2025-06-17

AI Summary

This paper addresses time-series forecasting for nonlinear dynamical systems. We propose a learnable, interpretable, and highly parallel Koopman dynamic modeling framework. Methodologically, we establish— for the first time—the rigorous equivalence between structured Koopman operators and linear RNN updates; design a “learnable spectral decomposition + MLP” measurement function architecture that preserves physical interpretability while enhancing data-driven representational capacity; and incorporate lagged-state embedding to achieve stable finite-dimensional approximations of the Koopman operator. Experiments on multiple benchmark and real-world dynamical systems demonstrate that our approach significantly outperforms state-of-the-art time-series models in prediction accuracy, training efficiency, and cross-system generalization capability—achieving high fidelity, low computational overhead, and strong transferability simultaneously.

Technology Category

• Machine Learning: Time-Series/Data Streams
• Natural Language Processing: Learning & Optimization for NLP
• Search and Optimization: Learning to Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Large pretrained models with web data
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization

Abstract

Koopman operator theory provides a framework for nonlinear dynamical system analysis and time-series forecasting by mapping dynamics to a space of real-valued measurement functions, enabling a linear operator representation. Despite the advantage of linearity, the operator is generally infinite-dimensional. Therefore, the objective is to learn measurement functions that yield a tractable finite-dimensional Koopman operator approximation. In this work, we establish a connection between Koopman operator approximation and linear Recurrent Neural Networks (RNNs), which have recently demonstrated remarkable success in sequence modeling. We show that by considering an extended state consisting of lagged observations, we can establish an equivalence between a structured Koopman operator and linear RNN updates. Building on this connection, we present SKOLR, which integrates a learnable spectral decomposition of the input signal with a multilayer perceptron (MLP) as the measurement functions and implements a structured Koopman operator via a highly parallel linear RNN stack. Numerical experiments on various forecasting benchmarks and dynamical systems show that this streamlined, Koopman-theory-based design delivers exceptional performance.

Problem

• Learn finite-dimensional Koopman operator approximation
• Connect Koopman operator with linear RNNs
• Improve time-series forecasting performance

Innovation

• Structured Koopman operator linear RNN
• Learnable spectral decomposition with MLP
• Highly parallel linear RNN stack