变分模拟算子网络
Variationally Mimetic Operator Networks
AI总结：
本文提出变分模拟算子网络（VarMiON），通过模仿变分弱解形式构建算子网络，在参数相近时比DeepONet和MIONet误差更小且更鲁棒。
摘要：
近年来，算子网络作为求解偏微分方程（PDEs）的深度学习工具崭露头角。这些网络将描述材料属性、强迫函数和边界数据的输入函数映射到偏微分方程的解。本文描述了一种新的算子网络架构，它模仿从问题的近似变分或弱形式获得的数值解的形式。将这些思想应用于一般的椭圆型偏微分方程，产生了变分模拟算子网络（VarMiON）。与传统的深度算子网络（DeepONet）类似，VarMiON 也由一个构建输出基函数的子网络和另一个构建这些基函数系数的子网络组成。然而，与 DeepONet 相比，VarMiON 中这些子网络的架构是精确确定的。对 VarMiON 解的误差分析表明，其误差包含来自训练数据的误差、训练误差、采样输入和输出函数时的求积误差，以及一个“覆盖误差”，该误差衡量测试输入函数与训练数据集中最近函数之间的距离。它还依赖于精确解算子及其 VarMiON 近似的稳定性常数。将 VarMiON 应用于一个典型的椭圆型偏微分方程和一个非线性偏微分方程，结果表明，在网络参数数量大致相同的情况下，VarMiON 平均比标准 DeepONet 和最近提出的多输入算子网络（MIONet）产生更小的误差。此外，其性能对输入函数的变化、用于采样输入和输出函数的技术、用于构建基函数的技术以及输入函数的数量更具鲁棒性。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
