Kolmogorov and Arnold, in answering Hilbert's 13th problem (in the context of continuous functions), laid the foundations for the modern theory of Neural Networks (NNs). Their proof divides the representation of a multivariate function into two steps: The first (non-linear) inter-layer map gives a universal embedding of the data manifold into a single hidden layer whose image is patterned in such a way that a subsequent dynamic can then be defined to solve for the second inter-layer map. I interpret this pattern as "minor concentration" of the almost everywhere defined Jacobians of the interlayer map. Minor concentration amounts to sparsity for higher exterior powers of the Jacobians. We present a conceptual argument for how such sparsity may set the stage for the emergence of successively higher order concepts in today's deep NNs and suggest two classes of experiments to test this hypothesis.
Kolmogorov Arnold Informed neural network: A physics-informed deep learning framework for solving forward and inverse problems based on Kolmogorov Arnold Networks
Tsinghua University（清华大学）；Bauhaus-Universität Weimar（魏玛包豪斯大学）；CNPC Engineering Technology RD Company Limited（中国石油集团工程技术研究院有限公司）；Queensland University of Technology（昆士兰科技大学）；Leibniz University Hannover（汉诺威莱布尼茨大学）
Physics-Informed Kolmogorov-Arnold Networks for multi-material elasticity problems in electronic packaging
Beijing University of Technology（北京工业大学）；Dalian University of Technology（大连理工大学）；Leibniz University Hannover（汉诺威莱布尼茨大学）；Tongji University（同济大学）；Bauhaus University Weimar（魏玛包豪斯大学）
A Practitioner's Guide to Kolmogorov-Arnold Networks
Department of Mathematics, Hong Kong Baptist University（香港 Baptist大学数学系）；Institution for Foundations of Data Science, Yale University（数据科学基础研究所，耶鲁大学）；Robotics and Technology of Computers Lab., Universidad de Sevilla（机器人与计算机技术实验室，塞维利亚大学）
CT-PIKAN: Coordinate-Transformed Physics-Informed Kolmogorov-Arnold Network with Autograd-Based Metric Evaluation for Solving PDEs in Curvilinear Domains
Department of Aerospace Engineering, Sharif University of Technology（谢里夫理工大学航空航天工程系）
Sinc Kolmogorov-Arnold network and its application for solving PDEs with singularities
Skolkovo Institute of Science and Technology（斯克洛夫科学与技术研究所）；Southern University of Science and Technology（南方科技大学）；International Center for Mathematics（国际数学中心）；National Center for Applied Mathematics Shenzhen (NCAMS)（深圳应用数学中心）
