The concept of the path-dependent partial differential equation (PPDE) was
first introduced in the context of path-dependent derivatives in financial
markets. Its semilinear form was later identified as a non-Markovian backward
stochastic differential equation (BSDE). Compared to the classical PDE, the
solution of a PPDE involves an infinite-dimensional spatial variable, making it
challenging to approximate, if not impossible. In this paper, we propose a
neural rough differential equation (NRDE)-based model to learn PPDEs, which
effectively encodes the path information through the log-signature feature
while capturing the fundamental dynamics. The proposed continuous-time model
for the PPDE solution offers the benefits of efficient memory usage and the
ability to scale with dimensionality. Several numerical experiments, provided
to validate the performance of the proposed model in comparison to the strong
baseline in the literature, are used to demonstrate its effectiveness.

本文提出了一种基于神经粗糙微分方程（NRDE）模型来学习路径相关部分微分方程（PPDE），该模型通过对数签名特征有效地编码路径信息并捕捉基本动态，在 PPDE 解的连续时间模型中提供了内存使用效率和处理维度扩展性的优点，并通过多个数值实验验证了其性能优越性。

基于神经 RDE 模型求解路径相关偏微分方程

A Neural RDE-based model for solving path-dependent PDEs

Traffic forecasting is one of the most popular spatio-temporal tasks in the
field of machine learning. A prevalent approach in the field is to combine
graph convolutional networks and recurrent neural networks for the
spatio-temporal processing. There has been fierce competition and many novel
methods have been proposed. In this paper, we present the method of
spatio-temporal graph neural rough differential equation (STG-NRDE). Neural
rough differential equations (NRDEs) are a breakthrough concept for processing
time-series data. Their main concept is to use the log-signature transform to
convert a time-series sample into a relatively shorter series of feature
vectors. We extend the concept and design two NRDEs: one for the temporal
processing and the other for the spatial processing. After that, we combine
them into a single framework. We conduct experiments with 6 benchmark datasets
and 21 baselines. STG-NRDE shows the best accuracy in all cases, outperforming
all those 21 baselines by non-trivial margins.

本研究介绍了一种名为 Spatio-Temporal Graph Neural Rough Differential Equation (STG-NRDE) 的方法，该方法将神经网络粗糙微分方程（NRDE）和图卷积网络结合起来，通过时间和空间数据进行交互，以预测交通流量，并在 6 个基准数据集和 21 个基线模型中实现了最佳准确性。