5 papers
Convergence of Implicit Gradient Descent for Training Two-Layer Physics-Informed Neural Networks
Xianliang Xu, Ting Du, Wang Kong +3
The optimization algorithms are crucial in training physics-informed neural networks (PINNs), as unsuitable methods may lead to poor solutions. Compared to the common gradient desc…
Convergence Analysis of Natural Gradient Descent for Over-parameterized Physics-Informed Neural Networks
Xianliang Xu, Ting Du, Wang Kong +3
In the context of over-parameterization, there is a line of work demonstrating that randomly initialized (stochastic) gradient descent (GD) converges to a globally optimal solution…
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…
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…
Convergence analysis of wide shallow neural operators within the framework of Neural Tangent Kernel
Xianliang Xu, Ye Li, Zhongyi Huang
Neural operators are aiming at approximating operators mapping between Banach spaces of functions, achieving much success in the field of scientific computing. Compared to certain…