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

We use deep feedforward artificial neural networks to approximate solutions of partial differential equations of advection and diffusion type in complex geometries. We derive analytical expressions of the gradients of the cost function with respect to the network parameters, as well as the gradient of the network itself with respect to the input, for arbitrarily deep networks. 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 quasi-Newton methods. We provide detailed examples on how to use deep feedforward neural networks as a basis for further work on deep neural network approximations to partial differential equations. We highlight the benefits of deep compared to shallow neural networks and other convergence enhancing techniques.