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This paper introduces the Neural Bilinear Dynamical Model (NBDM), which addresses the limitations of traditional linear models in time series forecasting by employing a bilinear latent dynamical formulation. By leveraging Koopman theory to elevate nonlinear dynamics into a higher-dimensional latent space, NBDM effectively captures complex behaviors while incorporating control inputs and compensating for approximation errors. Experimental results across five real-world datasets reveal that NBDM significantly outperforms existing baselines, especially in long-horizon forecasting scenarios with and without control inputs.
NBDM outperforms traditional models by effectively capturing nonlinear dynamics, achieving superior long-horizon forecasting accuracy even with missing control inputs.
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.
