Understand
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.