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Recently, several deep learning (DL) methods for approximatinghigh-dimensional partial differential equations (PDEs) have been proposed. Theinterest that these methods have generated in the literature is in large partdue to simulations which appear to demonstrate that such DL methods have thecapacity to overcome the curse of dimensionality (COD) for PDEs in the sensethat the number of computational operations they require to achieve a certainapproximation accuracy $\varepsilon\in(0,\infty)$ grows at most polynomially inthe PDE dimension $d\in\mathbb N$ and the reciprocal of $\varepsilon$. Whilethere is thus far no mathematical result that proves that one of such methodsis indeed capable of overcoming the COD, there are now a number of rigorousresults in the literature that show that deep neural networks (DNNs) have theexpressive power to approximate PDE solutions without the COD in the sense thatthe number of parameters used to describe the approximating DNN grows at mostpolynomially in both the PDE dimension $d\in\mathbb N$ and the reciprocal ofthe approximation accuracy $\varepsilon>0$. Roughly speaking, in the literatureit is has been proved for every $T>0$ that solutions $u_d\colon[0,T]\times\mathbb R^d\to \mathbb R$, $d\in\mathbb N$, of semilinear heat PDEswith Lipschitz continuous nonlinearities can be approximated by DNNs with ReLUactivation at the terminal time in the $L^2$-sense without the COD providedthat the initial value functions $\mathbb R^d\ni x\mapsto u_d(0,x)\in\mathbbR$, $d\in\mathbb N$, can be approximated by ReLU DNNs without the COD. It isthe key contribution of this work to generalize this result by establishingthis statement in the $L^p$-sense with $p\in(0,\infty)$ and by allowing theactivation function to be more general covering the ReLU, the leaky ReLU, andthe softplus activation functions as special cases.

Deep neural networks with ReLU, leaky ReLU, and softplus activation  provably overcome the curse of dimensionality for Kolmogorov partial  differential equations with Lipschitz nonlinearities in the $L^p$-sense