The neural Ordinary Differential Equation (ODE) model has shown success in learning complex continuous-time processes from observations on discrete time stamps. In this work, we consider the modeling and forecasting of time series data that are non-stationary and may have sharp changes like spikes. We propose an RNN-based model, called RNN-ODE-Adap, that uses a neural ODE to represent the time development of the hidden states, and we adaptively select time steps based on the steepness of changes of the data over time so as to train the model more efficiently for the "spike-like" time series. Theoretically, RNN-ODE-Adap yields provably a consistent estimation of the intensity function for the Hawkes-type time series data. We also provide an approximation analysis of the RNN-ODE model showing the benefit of adaptive steps. The proposed model is demonstrated to achieve higher prediction accuracy with reduced computational cost on simulated dynamic system data and point process data and on a real electrocardiography dataset.

翻译：神经常微分方程（ODE）模型在从离散时间戳观测中学习复杂连续时间过程方面已展现出成功。本文研究具有非平稳性且可能包含尖峰等剧烈变化的时间序列数据的建模与预测问题。我们提出一种基于RNN的模型——RNN-ODE-Adap，该模型利用神经ODE表示隐藏状态的时间演化，并根据数据随时间变化的陡峭程度自适应选择时间步长，从而更高效地训练"尖峰式"时间序列。理论上，RNN-ODE-Adap能够对霍克斯型时间序列数据的强度函数实现一致估计。我们还通过逼近分析表明自适应步长对RNN-ODE模型的益处。在仿真动态系统数据、点过程数据以及真实心电图数据集上的实验证明，该模型能以更低的计算成本实现更高的预测精度。