Recent research has used deep learning to develop partial differential equation (PDE) models in science and engineering. The functional form of the PDE is determined by a neural network, and the neural network parameters are calibrated to available data. Calibration of the embedded neural network can be performed by optimizing over the PDE. Motivated by these applications, we rigorously study the optimization of a class of linear elliptic PDEs with neural network terms. The neural network parameters in the PDE are optimized using gradient descent, where the gradient is evaluated using an adjoint PDE. As the number of parameters become large, the PDE and adjoint PDE converge to a non-local PDE system. Using this limit PDE system, we are able to prove convergence of the neural network-PDE to a global minimum during the optimization. Finally, we use this adjoint method to train a neural network model for an application in fluid mechanics, in which the neural network functions as a closure model for the Reynolds-averaged Navier--Stokes (RANS) equations. The RANS neural network model is trained on several datasets for turbulent channel flow and is evaluated out-of-sample at different Reynolds numbers.

翻译：近期研究利用深度学习开发科学与工程中的偏微分方程模型。该类偏微分方程的函数形式由神经网络决定，并通过可用数据校准神经网络参数。嵌入式神经网络的校准可通过在偏微分方程约束下进行优化实现。受这些应用启发，本文严格研究了一类含神经网络项的线性椭圆型偏微分方程的优化问题。我们采用梯度下降法优化偏微分方程中的神经网络参数，其中梯度通过伴随偏微分方程计算。随着参数数量增大，原偏微分方程与伴随偏微分方程收敛至一个非局部偏微分方程组。利用该极限偏微分方程组，我们能够证明优化过程中神经网络-偏微分方程系统收敛至全局最小值。最后，我们将此伴随方法应用于流体力学中的神经网络模型训练，使神经网络作为雷诺平均Navier-Stokes（RANS）方程的闭合模型。该RANS神经网络模型基于多个湍流槽道流数据集进行训练，并在不同雷诺数下进行了样本外评估。