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 and a nonlinear PDE reveals that for approximately the same number of network parameters, on average the VarMiON incurs smaller errors than a standard DeepONet and a recently proposed multiple-input operator network (MIONet). 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，算子网络作为近似偏微分方程（PDE）解的新型深度学习工具而崭露头角。这类网络将描述材料属性、力函数和边界条件的输入函数映射至PDE的解。本文提出一种新型算子网络架构，其结构模仿通过问题近似变分或弱形式获得的数值解形式。将该思想应用于通用椭圆型PDE，可构建变分拟构算子网络（VarMiON）。与传统深度算子网络（DeepONet）类似，VarMiON也由构建输出基函数的子网络和构建这些基函数系数的子网络组成。然而与DeepONet不同的是，VarMiON中子网络的架构具有精确的确定形式。对VarMiON解的误差分析表明，该误差包含训练数据误差、训练误差、输入输出函数采样的求积误差，以及衡量测试输入函数与训练数据集中最近函数间距离的"覆盖误差"；其误差还依赖于精确解算子及其VarMiON近似的稳定性常数。将VarMiON应用于标准椭圆型PDE和非线性PDE的案例表明，在近似相同网络参数数量的条件下，VarMiON的平均误差低于标准DeepONet及近期提出的多输入算子网络（MIONet）。此外，其对输入函数变化、输入输出函数采样技术、基函数构建技术及输入函数数量的鲁棒性均更为优越。