The vector field of a controlled differential equation (CDE) describes the relationship between a control path and the evolution of a solution path. Neural CDEs (NCDEs) treat time series data as observations from a control path, parameterise a CDE's vector field using a neural network, and use the solution path as a continuously evolving hidden state. As their formulation makes them robust to irregular sampling rates, NCDEs are a powerful approach for modelling real-world data. Building on neural rough differential equations (NRDEs), we introduce Log-NCDEs, a novel and effective method for training NCDEs. The core component of Log-NCDEs is the Log-ODE method, a tool from the study of rough paths for approximating a CDE's solution. On a range of multivariate time series classification benchmarks, Log-NCDEs are shown to achieve a higher average test set accuracy than NCDEs, NRDEs, and two state-of-the-art models, S5 and the linear recurrent unit.

翻译：控制微分方程（CDE）的向量场描述了控制路径与解路径演化之间的关系。神经CDE（NCDE）将时间序列数据视为控制路径的观测值，利用神经网络参数化CDE的向量场，并将解路径作为连续演化的隐藏状态。由于其公式表述对不规则采样率具有鲁棒性，NCDE成为建模真实世界数据的强大方法。基于神经粗糙微分方程（NRDE），我们提出了Log-NCDE——一种训练NCDE的新型有效方法。Log-NCDE的核心组件是对数-ODE方法，这是粗糙路径研究中用于近似CDE解的工具。在一系列多变量时间序列分类基准测试中，Log-NCDE在平均测试集准确率上优于NCDE、NRDE以及两个最先进模型S5和线性递归单元。