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.
Is the neural tangent kernel of PINNs deep learning general partial differential equations always convergent ?
Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）
The Stochastic Occupation Kernel (SOCK) Method for Learning Stochastic Differential Equations
Fariborz Maseeh Department of Mathematics and Statistics（法里博兹·马塞厄数学与统计学系）；Portland State University（波特兰州立大学）；Translational Genomics Research Institute（转化基因组研究研究所）
Neural Tangent Kernel of Neural Networks with Loss Informed by Differential Operators
Tsinghua University（清华大学）；Yau Mathematical Sciences Center, Tsinghua University（清华大学丘成桐数学科学中心）；Yanqi Lake Beijing Institute of Mathematical Sciences and Applications（雁栖湖北京数学科学与应用研究院）
Kernel-Based Testing for Single-Cell Differential Analysis
Nantes Université（南特大学）；Centrale Nantes（南特中央理工学院）；Laboratoire de Mathématiques Jean Leray（让·勒雷数学实验室）；CNRS（法国国家科学研究中心）；Université de Lyon（里昂大学）；Ecole Normale Supérieure de Lyon（里昂高等师范学校）；Université Claude Bernard Lyon 1（里昂第一大学）；Institut Curie（居里研究所）；PSL University（巴黎文理研究大学）
