超越线性动力学：用于时间序列预测的神经双线性动力学模型
Beyond Linear Dynamics: Neural Bilinear Dynamical Models for Time Series Forecasting
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中文总结 AI 辅助
针对时间序列预测中线性模型难以捕捉非线性动力学的问题，本文提出神经双线性动力学模型NBDM，结合Koopman理论与误差补偿项，设计增强记忆控制器处理控制缺失，在五组数据集上预测性能优于基线方法。
中文摘要 AI 辅助
实际应用中的时间序列通常由非线性动力学系统生成，这使得准确预测颇具挑战性。现有明确建模系统动力学的方法通常依赖线性假设或基于Koopman的线性化，可能无法充分捕捉复杂的非线性行为，并导致长期预测中的误差累积。为解决这一局限，我们提出神经双线性动力学模型（Neural Bilinear Dynamical Model，NBDM），该模型通过双线性隐式动力学公式对非线性系统动力学进行建模。具体而言，NBDM利用Koopman理论将原始非线性动力学提升至高维隐空间，在该空间中构建双线性动力学模型以表征状态演化。为缓解双线性表示引入的近似误差，我们进一步加入参数化误差补偿项。在该公式中，控制输入被明确整合至动力学中，当存在可用辅助变量时使用辅助变量，否则使用学习得到的反馈信号。为处理控制输入缺失的场景，我们设计了一种增强记忆的控制器，通过历史状态与控制信号间的乘法交互来推断隐式控制。在五个真实世界数据集上的实验表明，NBDM在给定控制输入和控制输入缺失的场景下，均始终优于竞争性基线方法，尤其在多步和长期预测中表现突出。
英文摘要
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.
发表机构
- Hangzhou Dianzi University（杭州电子科技大学）
机构由 AI 辅助整理，请以论文原文为准。
