标题： 多层神经网络求解高维参数障碍问题 显示英文标题

标题： Multi-level Neural Networks for high-dimensional parametric obstacle problems

摘要： 提出并分析了一种求解计算上具有挑战性的（随机）参数障碍问题的新方法，其中参数可能影响相关的偏微分方程（PDE），并决定障碍的位置和表面结构。 作为控制方程，假设了一个静止的椭圆扩散问题。障碍问题的高维解由一个专门构造的卷积神经网络（CNN）近似。这一新算法受到有限元约束多重网格算法的启发，用于表示参数到解的映射。这有两个好处：首先，由于适当的数据预处理，多级数据可以作为神经网络（NN）的显式输出，从而允许高效的实际计算。这提高了训练过程的有效性，并随后导致自然能量范数中的小误差。其次，CNN与多重网格算法的比较提供了对所提出的NN架构进行完整的先验收敛性和复杂性分析的方法。数值实验表明，该方法在这类具有挑战性的问题中表现出最先进的性能。 显示英文摘要

摘要： A new method to solve computationally challenging (random) parametric obstacle problems is developed and analyzed, where the parameters can influence the related partial differential equation (PDE) and determine the position and surface structure of the obstacle. As governing equation, a stationary elliptic diffusion problem is assumed. The high-dimensional solution of the obstacle problem is approximated by a specifically constructed convolutional neural network (CNN). This novel algorithm is inspired by a finite element constrained multigrid algorithm to represent the parameter to solution map. This has two benefits: First, it allows for efficient practical computations since multi-level data is used as an explicit output of the NN thanks to an appropriate data preprocessing. This improves the efficacy of the training process and subsequently leads to small errors in the natural energy norm. Second, the comparison of the CNN to a multigrid algorithm provides means to carry out a complete a priori convergence and complexity analysis of the proposed NN architecture. Numerical experiments illustrate a state-of-the-art performance for this challenging problem.