Solving Differential Equations using Neural Network

Neural Networks (NNs) in recent years have evolved as a framework to solve various complex mathematical equations. NN has numerous real life applications in almost every field like medicines, biometrics, automation industry, pharmaceutical etc. This has prompted mathematicians to explore NN technique to get the approximate solutions of many physical problems for which analytical solutions may not exist. In mathematical modelling, differential equations play a crucial role for solving the physical problems. In this paper, application of NN as universal solvers for Ordinary Differential Equations (ODEs) and Partial Differential Equations (PDEs) has been explained. The Neural Network is developed for initial and boundary value problems of different order ODEs and PDEs. The main objective here is to develop the single neural network for solving these equations. This is achieved by applying back-propagation learning algorithm with single layer and logistic sigmoid function. The results are tested for different number of neurons and various losses are generated to get the best loss corresponding to the output. The results show high degree of accuracy between the true solution and the approximate solution as much as the resulting error reaches 10−6 for ODEs and 10−3 for PDE. Further, the results exhibit that the same strategy developed here can be applied to solve any order PDEs and ODEs.