Deep neural networks (DNNs) with ReLU activation function are proved to be able to express viscosity solutions of linear partial integrodifferental equations (PIDEs) on state spaces of possibly high dimension $d$. Admissible PIDEs comprise Kolmogorov equations for high-dimensional diffusion, advection, and for pure jump L\'{e}vy processes. We prove for such PIDEs arising from a class of jump-diffusions on $\mathbb{R}^d$ that for any suitable measure $\mu^d$ on $\mathbb{R}^d$ there exist constants $C,{\mathfrak{p}},{\mathfrak{q}}>0$ such that for every $\varepsilon \in (0,1]$ and for every $d\in \mathbb{N}$ the DNN $L^2(\mu^d)$-expression error of viscosity solutions of the PIDE is of size $\varepsilon$ with DNN size bounded by $Cd^{\mathfrak{p}}\varepsilon^{-\mathfrak{q}}$. In particular, the constant $C>0$ is independent of $d\in \mathbb{N}$ and of $\varepsilon \in (0,1]$ and depends only on the coefficients in the PIDE and the measure used to quantify the error. This establishes that ReLU DNNs can break the curse of dimensionality (CoD for short) for viscosity solutions of linear, possibly degenerate PIDEs corresponding to suitable Markovian jump-diffusion processes. As a consequence of the employed techniques we also obtain that expectations of a large class of path-dependent functionals of the underlying jump-diffusion processes can be expressed without the CoD.

翻译：具有 ReLU 激活功能的深神经网络( DNN) 已被证明能够在可能高维的州空间中表达线性部分内分异方程式( PIDE) 的粘度解决方案。 允许 PIDE 包含高维扩散、 振荡和纯跳跃的 Kolmogorov 方程式 。 我们证明这种PIDE 产生于 $\ mathbb{ R ⁇ d$ 的跳- 扩散级 $\ mathbb{ 美元, 任何合适的度量 $\ mud$, $mathb{ R ⁇ d$ 的直线性部分内分异方方方方方方方方方程式( PIDE) 常数 $,\ mathfrak{q ⁇ 0, 等於每立方立方方方立方立方方立方方方方方方程式的數數數值數值數值 。