Simple recurrent neural networks for the numerical solutions of ODEs with Dirichlet boundary conditions

Korhan Günel, Gülsüm İşman, Merve Kocakula


In this study, we consider Dirichlet Boundary Value Problems (DBVPs) for Ordinary Differential Equations (ODEs) to illustrate the general procedure of obtaining numerical solutions using simple Recurrent Neural Networks (RNNs).  Different types of both linear and nonlinear activation functions are used in the neural network.  The network is trained by Particle Swarm Optimization (PSO) method, and cross validation approach is performed to tune the arbitrary parameters of neural nets.  The exact solutions and the obtained neural net solutions, regarding with the types of activation functions, are compared to determine the efficiency of using RNNs in solving the problem.  In all cases, the exact solutions are confronted with those obtained from RNNs in the context of absolute errors and average mean squared errors (MSEs) with standard deviations.

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