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

This repository provides the PyTorch implementation of:

Beyond Linear Dynamics: Neural Bilinear Dynamical Models for Time Series Forecasting
Mengzhou Gao, Huangqian Yu, Pengfei Jiao
Proceedings of the 32nd ACM SIGKDD Conference on Knowledge Discovery and Data Mining (KDD), 2026
Paper

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Introduction

Time series forecasting is a fundamental yet challenging problem due to the complex nonlinear dynamics of real-world systems, which are difficult to capture using linear or Koopman-based approximations.

In this work, we propose the Neural Bilinear Dynamical Model (NBDM), a novel framework that explicitly models nonlinear system dynamics through bilinear evolution in the Koopman latent space.

Our method:

• a bilinear dynamic propagation module to capture multiplicative interactions between latent system states and control inputs
• a learnable error compensation term to correct residual modeling errors arising from latent representations and bilinear approximations
• a memory-enhanced feedback controller to infer latent control signals from historical state-control interactions when external control inputs are unavailable.

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⚙️ Dependencies

NBDM is implemented in PyTorch with CUDA support. The required dependencies are:

numpy<2.0
torch==2.1.1+cu118
pandas
scipy
scikit-learn
matplotlib

To install PyTorch 2.1.1 with CUDA 11.8 support, run:

pip install torch==2.1.1+cu118 --index-url https://download.pytorch.org/whl/cu118

Then install the remaining dependencies:

pip install "numpy<2.0" pandas scipy scikit-learn matplotlib

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🚀 Quick Start

1. Data Preparation

Given Control Inputs

• PeMS04
• PeMS08

The PeMS04 and PeMS08 datasets are available at: https://github.com/guoshnBJTU/ASTGNN

Missing Control Inputs

• Temperature
• Seoul PM2.5
• PeMS-Bay

The Temperature and Seoul PM2.5 datasets are available at: https://github.com/QipengW/SF-GDE/

The PeMS-Bay dataset is available at: https://github.com/george-j-ste/Augmented-PEMS-BAY/tree/main

2. Run NBDM

To run NBDM on the PeMS04 dataset:

cd "Given control input"
python run.py --config ./config_file/PEMS04.conf

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📖 Citation

If you find this work useful in your research, please cite:

@inproceedings{gao2026beyond,
  title={Beyond Linear Dynamics: Neural Bilinear Dynamical Models for Time Series Forecasting},
  author={Gao, Mengzhou and Yu, Huangqian and Jiao, Pengfei},
  booktitle={Proceedings of the 32nd ACM SIGKDD Conference on Knowledge Discovery and Data Mining V. 2},
  pages={1228--1237},
  year={2026}
}