FlightKooba: A Fast Interpretable FTP Model

TL;DR

FlightKooba tackles the need for fast, interpretable time-series forecasting in flight trajectory prediction by analytically constructing Koopman operators from HiPPO Legendre coefficients $c_t$ and control-theoretic state-space relations, avoiding heavy parameter training. It unifies HiPPO compression, Koopman linearization, and state-space synthesis to produce a discretized, analytically specified dynamic system with state-space matrices $A,B$ and a linear readout, achieving orders-of-magnitude reductions in trainable parameters and training time while maintaining competitive accuracy on signals with periodic structure or underlying physical laws (where the model acts as a low-pass filter). The authors provide a theoretical account of the intrinsic low-pass behavior, clarifying suitable application domains and limitations for high-frequency or chaotic data. Empirically, FlightKooba demonstrates substantial efficiency gains across CAFUC, Lorenz, and other public datasets, offering a practical backbone for resource-constrained time-series forecasting and a pathway toward interpretable, physics-informed forecasting components.

Abstract

Flight trajectory prediction (FTP) and similar time series tasks typically require capturing smooth latent dynamics hidden within noisy signals. However, existing deep learning models face significant challenges of high computational cost and insufficient interpretability due to their complex black-box nature. This paper introduces FlightKooba, a novel modeling approach designed to extract such underlying dynamics analytically. Our framework uniquely integrates HiPPO theory, Koopman operator theory, and control theory. By leveraging Legendre polynomial bases, it constructs Koopman operators analytically, thereby avoiding large-scale parameter training. The method's core strengths lie in its exceptional computational efficiency and inherent interpretability. Experiments on multiple public datasets validate our design philosophy: for signals exhibiting strong periodicity or clear physical laws (e.g., in aviation, meteorology, and traffic flow), FlightKooba delivers competitive prediction accuracy while reducing trainable parameters by several orders of magnitude and achieving the fastest training speed. Furthermore, we analyze the model's theoretical boundaries, clarifying its inherent low-pass filtering characteristics that render it unsuitable for sequences dominated by high-frequency noise. In summary, FlightKooba offers a powerful, efficient, and interpretable new alternative for time series analysis, particularly in resource-constrained environments.

Figures (6)

• Figure 1: The structure of FlightKooba. FlightKooba consists of two components: the first component (HIPPO) computes the coefficients corresponding to the polynomial that fits the system's observation function; the second component (Kooba) generates the corresponding Koopman operator based on the results from the first component, which is used for subsequent prediction tasks. The specific matrices $M$, $N$, $A$, $B$, etc., shown in the figure will be detailed in the subsequent sections.
• Figure 2: Within the framework of flight trajectory prediction applications, FlightKooba computes details. The observation function serves as the state equation, represented by Legendre polynomials. The HIPPO method calculates the coefficients for each polynomial, which are then used to generate the Koopman operator. This operator can predict a sequence of future state changes.
• Figure 3: Prediction results for some dimensions on the test set.
• Figure 4: Prediction results for some dimensions on the test set.
• Figure 5: Illustration of model limitations on high-frequency datasets. Left: Noise signal from CAFUC (Feature 6), where predictions collapse to the signal mean. Right: Model fails to capture dynamics in ETTh2 (Feature 1), consistent with its low-pass filtering characteristics.