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

TL;DR

This paper introduces SKOLR, a novel structured Koopman operator-based linear RNN framework for time-series forecasting, leveraging spectral decomposition and neural networks to improve prediction accuracy on complex dynamical systems.

Contribution

It establishes a connection between Koopman operator approximation and linear RNNs, and develops SKOLR, a new model integrating spectral decomposition with neural measurement functions.

Findings

Outperforms existing methods on multiple forecasting benchmarks.

Effectively models nonlinear dynamical systems with linear RNNs.

Demonstrates the advantage of Koopman-based structured RNNs in time-series prediction.

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.