arXiv subjects

Xianlin Jin

Publications and source records attributed to Xianlin Jin.

4 recordsLinked to original sources

arXiv2024

Gauss Newton method for solving variational problems of PDEs with neural network discretizaitons

The numerical solution of differential equations using machine learning-based approaches has gained significant popularity. Neural network-based discretization has emerged as a powerful tool for solving differential equations by parameterizing a set of functions. Various approaches, such as the deep Ritz method and physics-informed neural networks, have been developed for numerical solutions. Training algorithms, including gradient descent and greedy algorithms, have been proposed to solve the resulting optimization problems. In this paper, we focus on the variational formulation of the problem and propose a Gauss- Newton method for computing the numerical solution. We provide a comprehensive analysis of the superlinear convergence properties of this method, along with a discussion on semi-regular zeros of the vanishing gradient. Numerical examples are presented to demonstrate the efficiency of the proposed Gauss-Newton method.

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arXiv2023

Greedy Training Algorithms for Neural Networks and Applications to PDEs

Recently, neural networks have been widely applied for solving partial differential equations (PDEs). Although such methods have been proven remarkably successful on practical engineering problems, they have not been shown, theoretically or empirically, to converge to the underlying PDE solution with arbitrarily high accuracy. The primary difficulty lies in solving the highly non-convex optimization problems resulting from the neural network discretization, which are difficult to treat both theoretically and practically. It is our goal in this work to take a step toward remedying this. For this purpose, we develop a novel greedy training algorithm for shallow neural networks. Our method is applicable to both the variational formulation of the PDE and also to the residual minimization formulation pioneered by physics informed neural networks (PINNs). We analyze the method and obtain a priori error bounds when solving PDEs from the function class defined by shallow networks, which rigorously establishes the convergence of the method as the network size increases. Finally, we test the algorithm on several benchmark examples, including high dimensional PDEs, to confirm the theoretical convergence rate. Although the method is expensive relative to traditional approaches such as finite element methods, we view this work as a proof of concept for neural network-based methods, which shows that numerical methods based upon neural networks can be shown to rigorously converge.

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arXiv2023

Two families of $n$-rectangle nonconforming finite elements for sixth-order elliptic equations

In this paper, we propose two families of nonconforming finite elements on $n$-rectangle meshes of any dimension to solve the sixth-order elliptic equations. The unisolvent property and the approximation ability of the new finite element spaces are established. A new mechanism, called the exchange of sub-rectangles, for investigating the weak continuities of the proposed elements is discovered. With the help of some conforming relatives for the $H^3$ problems, we establish the quasi-optimal error estimate for the tri-harmonic equation in the broken $H^3$ norm of any dimension. The theoretical results are validated further by the numerical tests in both 2D and 3D situations.

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arXiv2021

A conservative difference scheme with optimal pointwise error estimates for two-dimensional space fractional nonlinear Schrödinger equations

In this paper, a linearized semi-implicit finite difference scheme is proposed for solving the two-dimensional (2D) space fractional nonlinear Schrödinger equation (SFNSE).The scheme has the property of mass and energy conservation on the discrete level, with an unconditional stability and a second order accuracy for both time and spatial variables. The main contribution of this paper is an optimal pointwise error estimate for the 2D SFNSE, which is rigorously established and proved for the first time. Moreover, a novel technique is proposed for dealing with the nonlinear term in the equation, which plays an essential role in the error estimation. Finally, the numerical results confirm well with the theoretical findings.

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