In recent years operator networks have emerged as promising deep learning tools for approximating the solution to partial differential equations (PDEs). These networks map input functions that describe material properties, forcing functions and boundary data to the solution of a PDE. This work describes a new architecture for operator networks that mimics the form of the numerical solution obtained from an approximate variational or weak formulation of the problem. The application of these ideas to a generic elliptic PDE leads to a variationally mimetic operator network (VarMiON). Like the conventional Deep Operator Network (DeepONet) the VarMiON is also composed of a sub-network that constructs the basis functions for the output and another that constructs the coefficients for these basis functions. However, in contrast to the DeepONet, the architecture of these sub-networks in the VarMiON is precisely determined. An analysis of the error in the VarMiON solution reveals that it contains contributions from the error in the training data, the training error, the quadrature error in sampling input and output functions, and a "covering error" that measures the distance between the test input functions and the nearest functions in the training dataset. It also depends on the stability constants for the exact solution operator and its VarMiON approximation. The application of the VarMiON to a canonical elliptic PDE reveals that for approximately the same number of network parameters, on average the VarMiON incurs smaller errors than a standard DeepONet. Further, its performance is more robust to variations in input functions, the techniques used to sample the input and output functions, the techniques used to construct the basis functions, and the number of input functions.
翻译:近年来,运营商网络已成为有希望的深层次学习工具,以接近部分差异方程式(PDE)的解决方案。这些网络还绘制了描述物质属性、强制功能和边界数据以找到PDE的解决方案的输入功能。这项工作描述了运营商网络的新架构,它模仿了从问题大致变异或微弱的配方中获得的数字解决方案的形式。将这些想法应用到通用的椭圆式 PDE 操作器网络(VarmiON) 中,这与常规深海运营商网络(DeepONet) 网络(DeepONet) 一样,它也由一个子网络组成,该子网络为输出构建了基础功能的基础功能,该子网络构建了基础功能的基值。然而,与DeepONet相比,VarMiON的这些子网络结构结构得到了准确的确定。对Varmion解决方案的错误分析表明,它包含来自培训数据、培训错误、输入和输出函数的深度错误,以及“隐藏错误”,它用来测量试算数据功能之间的距离。它用来计算常态输入技术的运行稳定性,它也决定了它所使用的常态输入功能。