In this paper, we study adaptive neuron enhancement (ANE) method for solving self-adjoint second-order elliptic partial differential equations (PDEs). The ANE method is a self-adaptive method generating a two-layer spline NN and a numerical integration mesh such that the approximation accuracy is within the prescribed tolerance. Moreover, the ANE method provides a natural process for obtaining a good initialization which is crucial for training nonlinear optimization problem.
The underlying PDE is discretized by the Ritz method using a two-layer spline neural network based on either the primal or dual formulations that minimize the respective energy or complimentary functionals. Essential boundary conditions are imposed weakly through the functionals with proper norms. It is proved that the Ritz approximation is the best approximation in the energy norm; moreover, effect of numerical integration for the Ritz approximation is analyzed as well. Two estimators for adaptive neuron enhancement method are introduced, one is the so-called recovery estimator and the other is the least-squares estimator. Finally, numerical results for diffusion problems with either corner or intersecting interface singularities are presented.
An adaptive Deep Ritz framework for second-order fully nonlinear partial differential equations
Geneva School of Business Administration (HEG-Genève), University of Applied Sciences and Arts Western Switzerland (HES-SO)（日内瓦商业管理学院（HEG-Genève），瑞士西部应用科学和艺术大学（HES-SO））；Institute of Mathematics, Ecole Polytechnique Fédérale de Lausanne（洛桑联邦理工学院数学研究所）
RUNNs: Ritz-Uzawa Neural Networks for Solving Variational Problems
Basque Center for Applied Mathematics（巴斯克应用数学中心）；Instituto de Matemáticas, Pontificia Universidad Católica de Valparaíso（天主教法科大学瓦帕莱索数学研究所）；Department of Mathematics, CUNEF Universidad（CUNEF大学数学系）；Curtin University（科廷大学）；Department of Mathematics, University of the Basque Country UPV/EHU（巴斯克国立大学数学系）；Ikerbasque: Basque Foundation for Science（伊克拉巴斯克：巴斯克科学基金会）；School of Mathematical Sciences, University of Nottingham（诺丁汉大学数学科学学院）
Deep neural networks with ReLU, leaky ReLU, and softplus activation provably overcome the curse of dimensionality for space-time solutions of semilinear partial differential equations
University of Wuppertal（伍珀塔尔大学）；The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））；University of Münster（明斯特大学）；Toyota Financial Services（丰田金融服务公司）
Multilevel Picard approximations and deep neural networks with ReLU, leaky ReLU, and softplus activation overcome the curse of dimensionality when approximating semilinear parabolic partial differential equations in $L^p$-sense
Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore（数学科学系，物理与数学科学学院，南洋理工大学，新加坡）；Faculty of Mathematics, Bielefeld University, Germany（数学学院，比勒菲尔德大学，德国）
Deep neural networks with ReLU, leaky ReLU, and softplus activation provably overcome the curse of dimensionality for Kolmogorov partial differential equations with Lipschitz nonlinearities in the $L^p$-sense
Department of Mathematics & Informatics, University of Wuppertal（乌尔姆大学数学与信息学系）；School of Data Science and Shenzhen Research Institute of Big Data, The Chinese University of Hong Kong, Shenzhen (CUHK-Shenzhen)（香港中文大学（深圳）数据科学学院及深圳大数据研究院）；Applied Mathematics: Institute for Analysis and Numerics, University of Münster（穆恩斯特大学应用数学系及分析与数值分析研究所）；Department of Mathematical Sciences, University of Arkansas（阿肯色大学数学科学系）；Center for Astrophysics, Space Physics, and Engineering Research, Baylor University（贝勒大学天体物理学、空间物理学与工程研究中心）
Deflation-PINNs: Learning Multiple Solutions for PDEs and Landau-de Gennes
Department of Mathematics, Indian Institute of Science, Bangalore（印度科学研究所数学系，班加罗尔）；Department of Computing and Mathematical Sciences, California Institute of Technology（加州理工学院计算与数学科学系）
An $r$-adaptive finite element method using neural networks for parametric self-adjoint elliptic problem
Faculty of Engineering, Pontificia Universidad Católica de Chile（智利天主教大学工程学院）；Institute for Mathematical and Computational Engineering (IMC), Pontificia Universidad Católica de Chile（智利天主教大学数学与计算工程研究所）；University of the Basque Country (UPV/EHU)（巴斯克地区大学）；Basque Center for Applied Mathematics (BCAM)（巴斯克应用数学中心）
Quantifying Training Difficulty and Accelerating Convergence in Neural Network-Based PDE Solvers
Hong Kong University of Science and Technology（香港科技大学）；Shanghai Jiao Tong University（上海交通大学）；The Pennsylvania State University（宾夕法尼亚州立大学）；HKUST Shenzhen-Hong Kong Collaborative Innovation Research Institute（香港科技大学深港协同创新研究院）；Shanghai Artificial Intelligence Laboratory（上海人工智能实验室）
Expressive Power of Deep Networks on Manifolds: Simultaneous Approximation
School of Mathematical Sciences, Fudan University（复旦大学数学学院）；Fudan University（复旦大学）；Shanghai Key Laboratory for Contemporary Applied Mathematics, Fudan University（复旦大学当代应用数学重点实验室）
