Stochastic Volterra equations (SVEs) serve as mathematical models for the time evolutions of random systems with memory effects and irregular behaviour. We introduce neural stochastic Volterra equations as a physics-inspired architecture, generalizing the class of neural stochastic differential equations, and provide some theoretical foundation. Numerical experiments on various SVEs, like the disturbed pendulum equation, the generalized Ornstein--Uhlenbeck process and the rough Heston model are presented, comparing the performance of neural SVEs, neural SDEs and Deep Operator Networks (DeepONets).

翻译：随机Volterra方程（SVEs）是描述具有记忆效应和不规则行为的随机系统时间演化的数学模型。我们提出神经随机Volterra方程作为一种受物理学启发的架构，该架构推广了神经随机微分方程类别，并提供了相应的理论基础。本文展示了在多种SVE（如受扰单摆方程、广义Ornstein-Uhlenbeck过程及粗糙Heston模型）上的数值实验，比较了神经SVE、神经SDE与深度算子网络（DeepONets）的性能表现。