标题： 神经算子：偏微分方程的图核网络 显示英文标题

标题： Neural Operator: Graph Kernel Network for Partial Differential Equations

摘要： 神经网络的经典发展主要用于有限维欧几里得空间与一组类别之间，或者两个有限维欧几里得空间之间的映射。 本工作的目的是推广神经网络，使其能够学习无限维空间（算子）之间的映射。 我们工作的关键创新在于，在一个精心设计的网络架构中，一组单一的网络参数可以用来描述无限维空间之间的映射以及这些空间的不同有限维近似之间的映射。 我们通过组合非线性激活函数和一类积分算子来对无限维映射进行近似。 核积分通过图网络上的消息传递进行计算。 这种方法具有重要的实际意义，我们将在输入数据到偏微分方程（PDE）及其解之间的映射背景下加以说明。 在此背景下，这样的学习网络可以在不同的PDE近似方法（如有限差分或有限元方法）之间进行泛化，并在对应于不同基础分辨率和离散化水平的近似之间进行泛化。 实验结果证实了所提出的图核网络确实具有所需的特性，并且与最先进的求解器相比表现出有竞争力的性能。 显示英文摘要

摘要： The classical development of neural networks has been primarily for mappings between a finite-dimensional Euclidean space and a set of classes, or between two finite-dimensional Euclidean spaces. The purpose of this work is to generalize neural networks so that they can learn mappings between infinite-dimensional spaces (operators). The key innovation in our work is that a single set of network parameters, within a carefully designed network architecture, may be used to describe mappings between infinite-dimensional spaces and between different finite-dimensional approximations of those spaces. We formulate approximation of the infinite-dimensional mapping by composing nonlinear activation functions and a class of integral operators. The kernel integration is computed by message passing on graph networks. This approach has substantial practical consequences which we will illustrate in the context of mappings between input data to partial differential equations (PDEs) and their solutions. In this context, such learned networks can generalize among different approximation methods for the PDE (such as finite difference or finite element methods) and among approximations corresponding to different underlying levels of resolution and discretization. Experiments confirm that the proposed graph kernel network does have the desired properties and show competitive performance compared to the state of the art solvers.