Beyond Linear Dynamics: Neural Bilinear Dynamical Models for Time Series Forecasting

2026-08-05

AI Summary

This work addresses the limitations of existing nonlinear dynamical system modeling approaches, which often rely on linear assumptions or Koopman-based linearization and consequently struggle to accurately capture complex dynamics, leading to error accumulation in long-horizon predictions. To overcome this, the paper proposes the Neural Bilinear Dynamical Model (NBDM), which uniquely integrates Koopman theory with a bilinear dynamical structure to model state evolution in a high-dimensional latent space. NBDM further incorporates a parameterized error compensation mechanism to enhance predictive accuracy. To handle missing control inputs, a memory-augmented controller is introduced to infer implicit control signals. Experiments on five real-world datasets demonstrate that NBDM consistently outperforms baseline methods—regardless of whether control inputs are known—and achieves particularly strong performance in multi-step and long-horizon prediction tasks.

Abstract

Time series in real-world applications are often generated by nonlinear dynamical systems, making accurate forecasting challenging. Existing approaches that explicitly model system dynamics typically rely on linear assumptions or Koopman-based linearizations, which may inadequately capture complex nonlinear behaviors and lead to error accumulation in long-horizon prediction. To address this limitation, we propose the Neural Bilinear Dynamical Model (NBDM), which models nonlinear system dynamics through a bilinear latent dynamical formulation. Specifically, NBDM leverages Koopman theory to lift the original nonlinear dynamics into a higher-dimensional latent space, where a bilinear dynamical model is constructed to characterize state evolution. To mitigate the approximation error introduced by bilinear representations, we further incorporate a parameterized error compensation term. Within this formulation, control inputs are explicitly integrated into the dynamics, using auxiliary variables when available and learned feedback signals otherwise. To handle scenarios with missing control inputs, we design a memory-enhanced controller that infers latent controls through multiplicative interactions between historical states and control signals. Experiments on five real-world datasets demonstrate that NBDM consistently outperforms competitive baselines in both given-control and missing-control settings, particularly for multi-step and long-horizon forecasting.

Problem

• nonlinear dynamics
• time series forecasting
• bilinear models
• Koopman theory
• long-horizon prediction

Innovation

• Neural Bilinear Dynamical Model
• Koopman theory
• nonlinear time series forecasting
• latent control inference
• error compensation