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math.NA2024
Physics-Informed Tailored Finite Point Operator Network for Parametric Interface Problems
Ting Du, Xianliang Xu, Wang Kong +2
Learning operators for parametric partial differential equations (PDEs) using neural networks has gained significant attention in recent years. However, standard approaches like De…
math.NA2024
A Priori Estimation of the Approximation, Optimization and Generalization Errors of Random Neural Networks for Solving Partial Differential Equations
Xianliang Xu, Ye Li, Zhongyi Huang
In recent years, neural networks have achieved remarkable progress in various fields and have also drawn much attention in applying them on scientific problems. A line of methods i…
math.NA2024
Refined generalization analysis of the Deep Ritz Method and Physics-Informed Neural Networks
Xianliang Xu, Ye Li, Zhongyi Huang
In this paper, we derive refined generalization bounds for the Deep Ritz Method (DRM) and Physics-Informed Neural Networks (PINNs). For the DRM, we focus on two prototype elliptic…