We combine concepts from multilevel solvers for partial differential equations (PDEs) with neural network based deep learning and propose a new methodology for the efficient numerical solution of high-dimensional parametric PDEs. An in-depth theoretical analysis shows that the proposed architecture is able to approximate multigrid V-cycles to arbitrary precision with the number of weights only depending logarithmically on the resolution of the finest mesh. As a consequence, approximation bounds for the solution of parametric PDEs by neural networks that are independent on the (stochastic) parameter dimension can be derived. The performance of the proposed method is illustrated on high-dimensional parametric linear elliptic PDEs that are common benchmark problems in uncertainty quantification. We find substantial improvements over state-of-the-art deep learning-based solvers. As particularly challenging examples, random conductivity with high-dimensional non-affine Gaussian fields in 100 parameter dimensions and a random cookie problem are examined. Due to the multilevel structure of our method, the amount of training samples can be reduced on finer levels, hence significantly lowering the generation time for training data and the training time of our method.

翻译：我们将偏微分方程（PDE）的多层级求解器概念与基于神经网络的深度学习相结合，提出了一种高效数值求解高维参数化PDE的新方法。深入的理论分析表明，所提出的架构能够以任意精度逼近多重网格V-cycle，且权重数量仅随最细网格分辨率的对数增长。由此，可以推导出神经网络求解参数化PDE的逼近误差界，该误差界与（随机）参数维度无关。本文以不确定性量化中常见的基准问题——高维参数化线性椭圆型PDE为例，展示了所提方法的性能。我们发现，该方法相较于最先进的基于深度学习的求解器有显著改进。作为特别具有挑战性的例子，我们研究了100维参数空间中具有高维非仿射高斯场的随机传导问题以及随机cookie问题。由于方法的多层级结构，较细层级上的训练样本数量得以减少，从而大幅降低了训练数据的生成时间以及方法的训练时间。