Numerical experiments indicate that deep learning algorithms overcome the curse of dimensionality when approximating solutions of semilinear PDEs. For certain linear PDEs and semilinear PDEs with gradient-independent nonlinearities this has also been proved mathematically, i.e., it has been shown that the number of parameters of the approximating DNN increases at most polynomially in both the PDE dimension $d\in \mathbb{N}$ and the reciprocal of the prescribed accuracy $\epsilon\in (0,1)$. The main contribution of this paper is to rigorously prove for the first time that deep neural networks can also overcome the curse dimensionality in the approximation of a certain class of nonlinear PDEs with gradient-dependent nonlinearities.

翻译：数值实验表明，深度学习算法在逼近半线性偏微分方程的解时克服了维数灾难。对于某些线性偏微分方程以及具有梯度无关非线性的半线性偏微分方程，这一点也已在数学上得到证明，即已证明逼近所用深度神经网络的参数数量在偏微分方程维度$d\in\mathbb{N}$和预定精度$\epsilon\in(0,1)$的倒数上最多呈多项式增长。本文的主要贡献是首次严格证明深度神经网络在逼近某类具有梯度依赖非线性的非线性偏微分方程时，也能克服维数灾难。