A Novel Neural Network-Based Approach Comparable to High-Precision Finite Difference Methods

Abstract

Deep learning methods using neural networks for solving partial differential equations (PDEs) have emerged as a new paradigm. However, many of these methods approximate solutions by optimizing loss functions, often encountering convergence issues and accuracy limitations. In this paper, we propose a novel deep learning approach that leverages the expressive power of neural networks to generate basis functions. These basis functions are then used to create trial solutions, which are optimized using the least-squares method to solve for coefficients in a system of linear equations. This method integrates the strengths of streaming PINNs and the traditional least-squares method, offering both flexibility and a high accuracy. We conducted numerical experiments to compare our method with the results of high-order finite difference schemes and several commonly used neural network methods (PINNs, lbPINNs, ELMs, and PIELMs). Thanks to the mesh-less feature of the neural network, it is particularly effective for complex geometries. The numerical results demonstrate that our method significantly enhances the accuracy of deep learning in solving PDEs, achieving error levels comparable to high-accuracy finite difference methods.