A unified deep artificial neural network approach to partial differential equations in complex geometries

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In this paper we use deep feedforward artificial neural networks to approximate solutions to partial differential equations in complex geometries.

• We show how to modify the backpropagation algorithm to compute the partial derivatives of the network output with respect to the space variables which is needed to approximate the differential operator.
• The method is based on an ansatz for the solution which requires nothing but feedforward neural networks and an unconstrained gradient based optimization method such as gradient descent or a quasi-Newton method.
• We show an example where classical mesh based methods cannot be used and neural networks can be seen as an attractive alternative.