ANN-based methods for solving partial differential equations: a survey

Abstract

Traditionally, partial differential equation (PDE) problems are solved numerically through a discretization process. Iterative methods are then used to determine the algebraic system generated by this process. Recently, scientists have emerged artificial neural networks (ANNs), which solve PDE problems without a discretization process. Therefore, in view of the interest in developing ANN in solving PDEs, scientists investigated the variations of ANN which perform better than the classical discretization approaches. In this study, we discussed three methods for solving PDEs effectively, namely Pydens, NeuroDiffEq and Nangs methods. Pydens is the modified Deep Galerkin method (DGM) on the part of the approximate functions of PDEs. Then, NeuroDiffEq is the ANN model based on the trial analytical solution (TAS). Lastly, Nangs is the ANN-based method which uses the grid points for the training data. We compared the numerical results by solving the PDEs in terms of the accuracy and efficiency of the three methods. The results showed that NeuroDiffeq and Nangs have better performance in solving high-dimensional PDEs than the Pydens, while Pydens is only suitable for low-dimensional problems.

Keywords

• Artificial neural networks
• discretization
• Pydens method
• Neuroiffeq method
• Nangs module
• partial differential equations