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

arXiv: https://arxiv.org/abs/1711.06464 · HTML: https://mlsanity.com/paper/1711.06464/

## Key facts

- Cited by 122 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Nov 2017.

## Built on by (most cited first, 30 of 122)

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## Abstract (arXiv)

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.