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

TL;DR

This paper presents a neural network-based method for solving partial differential equations in complex geometries, demonstrating advantages over traditional mesh-based methods and emphasizing the benefits of deep neural networks.

Contribution

The paper introduces a novel approach using deep feedforward neural networks with modified backpropagation for PDE solutions in complex geometries, bypassing mesh requirements.

Findings

Neural networks can approximate PDE solutions where classical methods fail.

Deep networks outperform shallow ones in this context.

The method effectively computes derivatives needed for differential operators.

Abstract

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.