python gauss seidel method differential equations

Accordingly, the solutions achieved from the direct DNN prediction usually have inadequate numerical accuracy. However, solutions of the PDEs, particularly the nonlinear PDEs, do not always satisfy the regularity condition 19. According to the universal approximation theorem, the piecewise continuous objective function is a necessary condition for the reliability of DNN approximation 17, 18. The idea of utilizing DNN to achieve numerical solutions of PDEs brings certain potential risks as well. In the numerical simulation of engineering problems, DNN is utilized as a direct solver of the approximated system, and solutions are found with high efficiency 13– 16. 11, 12 employed neural networks to solve multiscale problems.

10 proposed a neural network-based indicator to correct the irregular solution in the discontinuous Galerkin scheme. 7, 8 Due to its high computational efficiency and scalability, especially on heterogeneous platforms, DNN has become a promising technique in scientific computing and even provides the possibility for real-time PDE solving 9. In recent years, deep neural network (DNN), which reveals superior capability to process and to predict complicated systems, has been widely employed in a variety of fields, e.g., natural language processing 5, computer vision 6, etc. Classic iterative solvers assure the precision and the reliability of the solutions but bring the challenge in terms of computational consumption. The discrete PDEs form sparse linear equations and are usually solved by iteration methods, e.g., the Gauss–Seidel method 1, the conjugate gradient (PCG) method, etc. The numerical methods for solving partial differential equations (PDEs) are among the most challenging and critical engineering problems.

Python gauss seidel method differential equations