A unified deep artificial neural network approach to partial differential equations in complex geometries
This provides a novel computational approach for researchers and engineers dealing with PDEs in domains where traditional methods are impractical, though it appears incremental in adapting neural networks to this domain.
The paper tackles solving partial differential equations in complex geometries using deep feedforward neural networks, demonstrating their viability as an alternative to classical mesh-based methods where those fail.
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. Finally, we highlight the benefits of deep compared to shallow neural networks and device some other convergence enhancing techniques.