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

arXiv: https://arxiv.org/abs/2608.04471 · HTML: https://mlsanity.com/paper/2608.04471/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 17 of its 38 references that are in the map.
- First posted on arXiv 5 Aug 2026.

## Builds on (most cited first, 17 of 17)

- [A Time Series is Worth 64 Words: Long-term Forecasting with Transformers](https://mlsanity.com/paper/2211.14730/) (2022)
- [Spatio-Temporal Graph Convolutional Networks: A Deep Learning Framework for Traffic Forecasting](https://mlsanity.com/paper/1709.04875/) (2017)
- [Graph WaveNet for Deep Spatial-Temporal Graph Modeling](https://mlsanity.com/paper/1906.00121/) (2019)
- [Connecting the Dots: Multivariate Time Series Forecasting with Graph Neural Networks](https://mlsanity.com/paper/2005.11650/) (2020)
- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [PDFormer: Propagation Delay-Aware Dynamic Long-Range Transformer for Traffic Flow Prediction](https://mlsanity.com/paper/2301.07945/) (2023)
- [Learning Deep Neural Network Representations for Koopman Operators of Nonlinear Dynamical Systems](https://mlsanity.com/paper/1708.06850/) (2017)
- [Forecasting Sequential Data using Consistent Koopman Autoencoders](https://mlsanity.com/paper/2003.02236/) (2020)
- [Koopa: Learning Non-stationary Time Series Dynamics with Koopman Predictors](https://mlsanity.com/paper/2305.18803/) (2023)
- [Multivariate Time Series Forecasting with Dynamic Graph Neural ODEs](https://mlsanity.com/paper/2202.08408/) (2022)
- [From Fourier to Koopman: Spectral Methods for Long-term Time Series Prediction](https://mlsanity.com/paper/2004.00574/) (2020)
- [Koopman Neural Forecaster for Time Series with Temporal Distribution Shifts](https://mlsanity.com/paper/2210.03675/) (2022)
- [Spatio-Temporal Meta Contrastive Learning](https://mlsanity.com/paper/2310.17678/) (2023)
- [An overview of Koopman-based control: From error bounds to closed-loop guarantees](https://mlsanity.com/paper/2509.02839/) (2025)
- [Signed Graph Neural Ordinary Differential Equation for Modeling Continuous-time Dynamics](https://mlsanity.com/paper/2312.11198/) (2023)
- [SafEDMD: A Koopman-based data-driven controller design framework for nonlinear dynamical systems](https://mlsanity.com/paper/2402.03145/) (2024)
- [Diffolio: A Diffusion Model for Multivariate Probabilistic Financial Time-Series Forecasting and Portfolio Construction](https://mlsanity.com/paper/2511.07014/) (2025)

## Abstract (arXiv)

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.