基于卷积神经网络的无网格形函数影响域优化研究

在无网格法中，离散节点之间的相互联系由节点形函数影响域的大小确定，因此形函数影响域的大小对无网格法的计算精度有着直接和重要的影响。但由于无网格形函数的形式较为复杂，目前形函数影响域大小的选择仍然缺乏系统的理论依据，通常在实际计算中仍凭借经验进行选取，难以保证计算精度。卷积神经网络是一类机器学习方法，其感受野与无网格形函数的影响域具有内在相似性，因此在形函数影响域选择方面有很好的适用性。基于该特性，本文通过引入卷积神经网络对无网格形函数的影响域进行了优化选择。首先，针对感受野和影响域的匹配关系，分析了卷积神经网络的结构设计和超参数选择，提出了一种无网格法内禀卷积神经网络结构的设计方法；然后依托该网络结构设计方法，建立了对无网格形函数影响域和数值解分别优化或同时优化的卷积神经网络。文中通过算例系统验证了所提无网格法内禀卷积神经网络对形函数影响域选择和计算结果的优化效应。

A CNN-based approach for optimizing support selection of meshfree methods

In meshfree methods, the discretized particles or nodes are inter-related by the supports of meshfree shape functions, and thus the support sizes of meshfree shape functions have a significant impact on the meshfree solution accuracy. However, due to the complexity of meshfree shape functions, currently there is still no general guidelines for meshfree support selection, and in practice the meshfree support size is often picked up by trial and error and a uniform support size is set for all particles which cannot ensure the meshfree solution accuracy. In this work, a machine learning approach, namely, the convolutional neural network (CNN), is introduced to optimize the meshfree support selection. It is shown that there is a strong similarity and linkage between the CNN receptive fields and meshfree supports, and accordingly a meshfree intrinsic CNN framework is proposed for meshfree analysis, where the network architecture and the corresponding determination of hyperparameters, such as the convolution kernel size, receptive field size, and their relationships with the number of meshfree nodes, the monomial basis function order and the support size, are particularly detailed for meshfree methods. Subsequently, within the proposed meshfree intrinsic CNN framework, several networks are designed to facilitate the meshfree support selection and solution prediction, in which a single objective like meshfree support selection, or multi-objectives like both meshfree support selection and solution prediction can be realized. Numerical results systematically demonstrate that the proposed meshfree intrinsic CNN framework is very effective to optimize the meshfree support selection with improved solution accuracy.