Efficient Shallow Ritz Method For 1D Diffusion Problems

Abstract

This paper studies the shallow Ritz method for solving the one-dimensional diffusion problem. It is shown that the shallow Ritz method improves the order of approximation dramatically for non-smooth problems. To realize this optimal or nearly optimal order of the shallow Ritz approximation, we develop a damped block Newton (dBN) method that alternates between updates of the linear and non-linear parameters. Per each iteration, the linear and the non-linear parameters are updated by exact inversion and one step of a modified, damped Newton method applied to a reduced non-linear system, respectively. The computational cost of each dBN iteration is $O(n)$. Starting with the non-linear parameters as a uniform partition of the interval, numerical experiments show that the dBN is capable of efficiently moving mesh points to nearly optimal locations. To improve efficiency of the dBN further, we propose an adaptive damped block Newton (AdBN) method by combining the dBN with the adaptive neuron enhancement (ANE) method [26].

References in corpus (6)

• Artificial Neural Networks for Solving Ordinary and Partial Differential Equations
• DGM: A deep learning algorithm for solving partial differential equations
• A unified deep artificial neural network approach to partial differential equations in complex geometries
• Deep least-squares methods: an unsupervised learning-based numerical method for solving elliptic PDEs
• Least-Squares ReLU Neural Network (LSNN) Method For Linear Advection-Reaction Equation
• Self-adaptive deep neural network: Numerical approximation to functions and PDEs