论文标题
kolmogorov通过深度学习方法在希尔伯特空间中的无限尺寸方程式
The Kolmogorov Infinite Dimensional Equation in a Hilbert space Via Deep Learning Methods
论文作者
论文摘要
我们考虑在希尔伯特太空中构成的非线性kolmogorov方程，不一定是有限的尺寸。 Cox等人最近研究了该模型。 [24]在随机波模型的弱收敛速率的框架中。在这里，我们通过提供无限维深度学习方法来近似该模型的合适解决方案来提出一种补充方法。基于Hure，Pham和Warin [45]关于有限尺寸案例的工作，以及我们先前处理基于Lévy的过程的工作[20]，我们将Euler方案和一致性结果推广到前向后的随机微分方程到Infinite dimenite Dimenite Hilbert估算的情况。由于我们的框架是一般的，因此我们需要最近开发的Deponets神经网络[21，51]来详细描述近似程序。同样，Fuhrman和Tessitore [35]开发的框架完全描述了随机近似值，将适应我们的设置
We consider the nonlinear Kolmogorov equation posed in a Hilbert space $H$, not necessarily of finite dimension. This model was recently studied by Cox et al. [24] in the framework of weak convergence rates of stochastic wave models. Here, we propose a complementary approach by providing an infinite-dimensional Deep Learning method to approximate suitable solutions of this model. Based in the work by Hure, Pham and Warin [45] concerning the finite dimensional case, and our previous work [20] dealing with Lévy based processes, we generalize an Euler scheme and consistency results for the Forward Backward Stochastic Differential Equations to the infinite dimensional Hilbert valued case. Since our framework is general, we require the recently developed DeepOnets neural networks [21, 51] to describe in detail the approximation procedure. Also, the framework developed by Fuhrman and Tessitore [35] to fully describe the stochastic approximations will be adapted to our setting
