SKOLR: Structured Koopman Operator Linear RNN for Time-Series Forecasting
This addresses forecasting challenges in domains like finance or climate, but is incremental as it builds on existing Koopman and RNN methods.
The paper tackles time-series forecasting by connecting Koopman operator theory to linear RNNs, proposing SKOLR, which integrates learnable spectral decomposition and MLPs to achieve exceptional performance on benchmarks and dynamical systems.
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.