KoNODE: Koopman-Driven Neural Ordinary Differential Equations with Evolving Parameters for Time Series Analysis

Neural ordinary differential equations (NODEs) have demonstrated strong capabilities in modeling time series. However, existing NODE- based methods often focus solely on the surface-level dynamics derived from observed states, which limits their ability to capture more complex underlying behaviors. To overcome this challenge, we propose KoNODE, a Koopman-driven NODE framework that explicitly models the evolution of ODE parameters over time to encode deep-level information. KoNODE captures the essential yet simple intrinsic linear dynamics that govern the surface dynamics by employing Koopman operators. Our framework operates at three hierarchical levels: the observed state dynamics, the parameter dynamics, and the Koopman linear dynamics, representing the fundamental driving rules of the state dynamics. The proposed approach offers significant improvements in two critical time series tasks: long-term prediction (enabled by the simple linear dynamics) and generalization to new data (driven by the evolving ODE parameters). We validate KoNODE through experiments on synthetic data from complex dynamic systems and real-world datasets, demonstrating its effectiveness in practical scenarios.

Understanding how things change over time — like weather patterns, stock markets, or heart rhythms — is a big challenge in science and technology. A popular approach is to use equations that model how these changes happen. But most current methods only look at what we can directly observe, often missing the hidden rules that actually drive the changes.Our work introduces a new method, called KoNODE, that digs deeper. Instead of just tracking what’s happening on the surface, KoNODE also models how the rules themselves evolve over time. It uses a mathematical tool called the Koopman operator, which helps uncover simple patterns beneath complex behaviors.This layered approach lets KoNODE make more accurate long-term predictions and adapt better when it sees new types of data — two big challenges in analyzing time-based information. We’ve tested KoNODE on both simulated systems and real-world data, and it consistently outperforms existing methods. We’ve also made our code freely available for others to use and build upon.