Solving Differential Equations by Means of Feed-Forward Artificial Neural Networks

A method for solving both, ordinary and partial, non-linear differential equations (DE) by means of the feed-forward artificial neural networks (ANN) is presented in this paper. Proposed approach consist in training ANN in such a way, that it approximates a function being a particular solution of DE and all its derivatives, up to the order of the equation. This is achieved by special construction of the cost function which contains informations about derivatives of the network. ANNs with sigmoidal activation functions in hidden nodes, thus infinitely differentiable, are considered in this paper. Illustrative examples of the solution of a non-linear DE are also presented.