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

Real-world time series often originate from complex nonlinear dynamical systems. While deep learning models like Transformers or RNNs are powerful, they often act as black boxes that lack explicit dynamical structure. Conversely, traditional Koopman-based methods use linear approximations in a lifted space, which can be too restrictive to capture the nuanced, multiplicative interactions between system states and external control inputs. This paper asks: can we design a more expressive, structured dynamical model that captures these nonlinear interactions while remaining computationally tractable?

The authors propose the Neural Bilinear Dynamical Model (NBDM). The framework operates in three main stages:

1. Bilinear Latent Dynamics: Using Koopman theory, the model lifts observations into a higher-dimensional latent space. Instead of enforcing strict linearity, it employs a bilinear formulation that explicitly models multiplicative interactions between latent states and control inputs.
2. Error Compensation: To account for the inevitable approximation errors inherent in finite-dimensional Koopman representations, the model includes a parameterized, learnable error term that captures residual dynamics.
3. Memory-Enhanced Controller: In scenarios where external control inputs are missing, the model uses a feedback mechanism that infers latent control signals by analyzing historical state-control interactions, allowing the model to adapt dynamically over time.

Experiments across five real-world datasets—covering climate, air quality, and traffic domains—demonstrate that NBDM consistently outperforms state-of-the-art baselines. The model shows particular strength in multi-step and long-horizon forecasting, where the explicit bilinear structure and error compensation mechanism prevent the error accumulation typically seen in purely linear or black-box approaches. The memory-enhanced controller also proved effective in scenarios where exogenous control data was unavailable, allowing the model to infer necessary latent drivers from historical patterns.