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.

控制微分方程的向量场描述了控制路径与解路径的演变之间的关系。神经 CDE（NCDEs）将时间序列数据视为来自控制路径的观测值，使用神经网络对 CDE 的向量场进行参数化，并将解路径用作持续演化的隐藏状态。通过引入 Log-NCDEs 方法，利用粗糙路径研究中的 Log-ODE 方法来近似 CDE 的解，Log-NCDEs 在多元时间序列分类基准测试中展现出比 NCDEs、NRDEs 和两种最先进模型 S5 和线性递归单元更高的平均测试集准确率。

基于对数神经控制微分方程的李括号效应

Log Neural Controlled Differential Equations: The Lie Brackets Make a  Difference

Machine learning (ML) holds great potential for accurately forecasting
treatment outcomes over time, which could ultimately enable the adoption of
more individualized treatment strategies in many practical applications.
However, a significant challenge that has been largely overlooked by the ML
literature on this topic is the presence of informative sampling in
observational data. When instances are observed irregularly over time, sampling
times are typically not random, but rather informative -- depending on the
instance's characteristics, past outcomes, and administered treatments. In this
work, we formalize informative sampling as a covariate shift problem and show
that it can prohibit accurate estimation of treatment outcomes if not properly
accounted for. To overcome this challenge, we present a general framework for
learning treatment outcomes in the presence of informative sampling using
inverse intensity-weighting, and propose a novel method, TESAR-CDE, that
instantiates this framework using Neural CDEs. Using a simulation environment
based on a clinical use case, we demonstrate the effectiveness of our approach
in learning under informative sampling.

本研究利用神经 CDEs 提出了一种基于所谓 “关键性采样” 的治疗结果预测框架，并证明在实践应用中该方法的有效性。

学习在时间上预测治疗结果时考虑信息采样

Accounting For Informative Sampling When Learning to Forecast Treatment  Outcomes Over Time

Hawkes processes are a popular framework to model the occurrence of
sequential events, i.e., occurrence dynamics, in several fields such as social
diffusion. In real-world scenarios, the inter-arrival time among events is
irregular. However, existing neural network-based Hawkes process models not
only i) fail to capture such complicated irregular dynamics, but also ii)
resort to heuristics to calculate the log-likelihood of events since they are
mostly based on neural networks designed for regular discrete inputs. To this
end, we present the concept of Hawkes process based on controlled differential
equations (HP-CDE), by adopting the neural controlled differential equation
(neural CDE) technology which is an analogue to continuous RNNs. Since HP-CDE
continuously reads data, i) irregular time-series datasets can be properly
treated preserving their uneven temporal spaces, and ii) the log-likelihood can
be exactly computed. Moreover, as both Hawkes processes and neural CDEs are
first developed to model complicated human behavioral dynamics, neural
CDE-based Hawkes processes are successful in modeling such occurrence dynamics.
In our experiments with 4 real-world datasets, our method outperforms existing
methods by non-trivial margins.

采用神经控制微分方程技术提出了基于控制微分方程的 Hawkes 进程（HP-CDE）概念，成功地对不规则时间序列数据集进行了建模，并能够准确地计算对数似然。在四个真实数据集上的实验中，该方法的表现优于现有方法。