It is a challenging topic in applied mathematics to solve high-dimensional nonlinear partial differential equations (PDEs). Standard approximation methods for nonlinear PDEs suffer under the curse of dimensionality (COD) in the sense that the number of computational operations of the approximation method grows at least exponentially in the PDE dimension and with such methods it is essentially impossible to approximately solve high-dimensional PDEs even when the fastest currently available computers are used. However, in the last years great progress has been made in this area of research through suitable deep learning (DL) based methods for PDEs in which deep neural networks (DNNs) are used to approximate solutions of PDEs. Despite the remarkable success of such DL methods in simulations, it remains a fundamental open problem of research to prove (or disprove) that such methods can overcome the COD in the approximation of PDEs. However, there are nowadays several partial error analysis results for DL methods for high-dimensional nonlinear PDEs in the literature which prove that DNNs can overcome the COD in the sense that the number of parameters of the approximating DNN grows at most polynomially in both the reciprocal of the prescribed approximation accuracy $\varepsilon>0$ and the PDE dimension $d\in\mathbb{N}$. In the main result of this article we prove that for all $T,p\in(0,\infty)$ it holds that solutions $u_d\colon[0,T]\times\mathbb{R}^d\to\mathbb{R}$, $d\in\mathbb{N}$, of semilinear heat equations with Lipschitz continuous nonlinearities can be approximated in the $L^p$-sense on space-time regions without the COD by DNNs with the rectified linear unit (ReLU), the leaky ReLU, or the softplus activation function. In previous articles similar results have been established not for space-time regions but for the solutions $u_d(T,\cdot)$, $d\in\mathbb{N}$, at the terminal time $T$.

翻译：求解高维非线性偏微分方程（PDE）是应用数学中的一个具有挑战性的课题。传统的非线性PDE近似方法受制于维数灾难（COD），即近似方法的计算操作数至少随PDE维数呈指数增长，因此即使使用当前最快的计算机，也基本上无法近似求解高维PDE。然而，近年来通过基于深度学习（DL）的PDE方法在这一研究领域取得了重大进展，其中深度神经网络（DNN）被用于近似PDE的解。尽管此类DL方法在模拟中取得了显著成功，但证明（或证伪）这些方法能否在PDE近似中克服COD仍然是基础研究中悬而未决的根本问题。不过，目前文献中已有若干关于高维非线性PDE的DL方法的部分误差分析结果，证明DNN能够克服COD，即近似DNN的参数数量在给定近似精度$\varepsilon>0$的倒数与PDE维数$d\in\mathbb{N}$上至多以多项式速度增长。本文的主要结果证明：对于所有$T,p\in(0,\infty)$，具有Lipschitz连续非线性的半线性热方程的解$u_d\colon[0,T]\times\mathbb{R}^d\to\mathbb{R}$（$d\in\mathbb{N}$）可在时空区域上被具有修正线性单元（ReLU）、泄漏修正线性单元（leaky ReLU）或softplus激活函数的DNN以$L^p$意义近似，且不产生COD。在先前的研究中，类似结果仅针对终端时刻$T$的解$u_d(T,\cdot)$（$d\in\mathbb{N}$）建立，而未涉及时空区域。