We study the problem of modeling a non-linear dynamical system when given a time series by deriving equations directly from the data. Despite the fact that time series data are given as input, models for dynamics and estimation algorithms that incorporate long-term temporal dependencies are largely absent from existing studies. In this paper, we introduce a latent state to allow time-dependent modeling and formulate this problem as a dynamics estimation problem in latent states. We face multiple technical challenges, including (1) modeling latent non-linear dynamics and (2) solving circular dependencies caused by the presence of latent states. To tackle these challenging problems, we propose a new method, Latent Non-Linear equation modeling (LaNoLem), that can model a latent non-linear dynamical system and a novel alternating minimization algorithm for effectively estimating latent states and model parameters. In addition, we introduce criteria to control model complexity without human intervention. Compared with the state-of-the-art model, LaNoLem achieves competitive performance for estimating dynamics while outperforming other methods in prediction.

翻译：本文研究如何直接从数据推导方程来建模非线性动力系统的问题。尽管输入数据为时间序列，现有研究大多缺乏能够融入长期时间依赖性的动力学模型及估计算法。本文引入潜在状态以实现时间依赖性建模，并将该问题形式化为潜在状态中的动力学估计问题。我们面临多个技术挑战，包括：(1) 建模潜在非线性动力学；(2) 解决因潜在状态存在而产生的循环依赖。为应对这些挑战，我们提出一种新方法——潜在非线性方程建模（LaNoLem），该方法能够建模潜在非线性动力系统，并提出一种新颖的交替最小化算法以有效估计潜在状态和模型参数。此外，我们引入了无需人工干预即可控制模型复杂度的准则。与最先进模型相比，LaNoLem在动力学估计方面取得具有竞争力的性能，同时在预测任务上优于其他方法。