We describe and analyze a hybrid finite element/neural network method for predicting solutions of partial differential equations. The methodology is designed for obtaining fine scale fluctuations from neural networks in a local manner. The network is capable of locally correcting a coarse finite element solution towards a fine solution taking the source term and the coarse approximation as input. Key observation is the dependency between quality of predictions and the size of training set which consists of different source terms and corresponding fine & coarse solutions. We provide the a priori error analysis of the method together with the stability analysis of the neural network. The numerical experiments confirm the capability of the network predicting fine finite element solutions. We also illustrate the generalization of the method to problems where test and training domains differ from each other.

翻译：我们描述并分析了一种用于预测偏微分方程解的混合有限元/神经网络方法。该方法旨在以局部方式从神经网络获取细尺度波动。该网络能够以源项和粗尺度近似作为输入，将粗尺度有限元解局部修正为细尺度解。关键观测在于预测质量与训练集大小之间的依赖关系，该训练集由不同的源项及对应的细/粗尺度解组成。我们提供了该方法的先验误差分析及神经网络的稳定性分析。数值实验证实了该网络预测细尺度有限元解的能力。我们还展示了该方法在测试域与训练域不同的问题上的泛化能力。