用卷积神经网络高效求解高维参数化障碍问题，精度高且可分析。
Multi-level Neural Networks for high-dimensional parametric obstacle problems
- 设计专用卷积网络，利用多层级数据预处理逼近参数到解的映射。
- 在自然能量范数下误差小，训练效率高，数值实验达领先水平。
- 适合需要高维参数化偏微分方程求解的研究者或工程应用。
本文提出一种新方法，用于求解计算困难的（随机）参数化障碍问题，其中参数可影响相关偏微分方程（PDE）并决定障碍物的位置与表面结构。假设控制方程为稳态椭圆型扩散问题。高维障碍问题的解通过一个专门构造的卷积神经网络（CNN）近似。该算法受有限元约束多网格算法启发，以表示参数到解的映射。其优势在于：首先，利用多层级数据作为网络显式输出，通过适当的数据预处理实现高效计算，提升训练效率，从而在自然能量范数下获得小误差；其次，通过将CNN与多网格算法对比，实现了对所提神经网络架构的完整先验收敛性与复杂度分析。数值实验表明，该方法在这一挑战性问题上达到当前最优性能。
原文摘要 · Abstract (English)
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.
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