To model time series accurately is important within a wide range of fields. As the world is generally too complex to be modelled exactly, it is often meaningful to assess the probability of a dynamical system to be in a specific state. This paper presents the Delay-SDE-net, a neural network model based on stochastic delay differential equations (SDDEs). The use of SDDEs with multiple delays as modelling framework makes it a suitable model for time series with memory effects, as it includes memory through previous states of the system. The stochastic part of the Delay-SDE-net provides a basis for estimating uncertainty in modelling, and is split into two neural networks to account for aleatoric and epistemic uncertainty. The uncertainty is provided instantly, making the model suitable for applications where time is sparse. We derive the theoretical error of the Delay-SDE-net and analyze the convergence rate numerically. At comparisons with similar models, the Delay-SDE-net has consistently the best performance, both in predicting time series values and uncertainties.

翻译：准确建模时间序列在许多领域至关重要。由于现实世界通常过于复杂而无法精确建模，评估动力系统处于特定状态的概率往往具有实际意义。本文提出延迟-随机微分方程网络（Delay-SDE-net），这是一种基于随机延迟微分方程（SDDEs）的神经网络模型。采用含多个延迟项的SDDEs作为建模框架，使其通过系统先前状态保留记忆效应，从而适用于具有记忆特性的时间序列。该网络的随机部分为建模不确定性估计提供了理论基础，并通过两个神经网络分别处理偶然不确定性与认知不确定性。模型可即时提供不确定性估计，适用于时间资源稀疏的应用场景。我们推导了Delay-SDE-net的理论误差并进行了收敛率的数值分析。在与同类模型的比较中，Delay-SDE-net在时间序列数值预测与不确定性估计方面均持续表现最优。