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.

本文利用深度前馈人工神经网络近似求解复杂几何下的偏微分方程，并演示了如何修改反向传播算法来计算网络输出对空间变量的偏导数。此方法基于一种假设解法，只需要前馈神经网络和梯度优化方法，如梯度下降或拟牛顿方法，可以作为网格法无法使用时的有效替代方案。此外，本文还阐述了深度相比于浅度神经网络的优势及其他收敛增强技术的设想。