School of Aeronautics and Astronautics, Sun Yat-sen University, Guangzhou, 510275, China.
Sci Rep. 2023 Mar 18;13(1):4479. doi: 10.1038/s41598-023-31236-0.
The consumption of solving large-scale linear equations is one of the most critical issues in numerical computation. An innovative method is introduced in this study to solve linear equations based on deep neural networks. To achieve a high accuracy, we employ the residual network architecture and the correction iteration inspired by the classic iteration methods. By solving the one-dimensional Burgers equation and the two-dimensional heat-conduction equation, the precision and effectiveness of the proposed method have been proven. Numerical results indicate that this DNN-based technique is capable of obtaining an error of less than 10. Moreover, its computation time is less sensitive to the problem size than that of classic iterative methods. Consequently, the proposed method possesses a significant efficiency advantage for large-scale problems.
求解大规模线性方程组的计算量是数值计算中最关键的问题之一。本研究提出了一种基于深度神经网络求解线性方程组的创新方法。为了达到高精度，我们采用了残差网络架构和经典迭代方法启发的校正迭代。通过求解一维 Burgers 方程和二维热传导方程，验证了所提出方法的精度和有效性。数值结果表明，这种基于 DNN 的技术能够获得小于 10 的误差。此外，与经典迭代方法相比，其计算时间对问题规模的敏感度较低。因此，所提出的方法在大规模问题上具有显著的效率优势。
