In this paper, we combine convolutional neural networks (CNNs) with reduced order modeling (ROM) for efficient simulations of multiscale problems. These problems are modeled by partial differential equations with high-dimensional random inputs. The proposed method involves two separate CNNs: Basis CNNs and Coefficient CNNs (Coef CNNs), which correspond to two main parts of ROM. The method is called CNN-based ROM. The former one learns input-specific basis functions from the snapshots of fine-scale solutions. An activation function, inspired by Galerkin projection, is utilized at the output layer to reconstruct fine-scale solutions from the basis functions. Numerical results show that the basis functions learned by the Basis CNNs resemble data, which help to significantly reduce the number of the basis functions. Moreover, CNN-based ROM is less sensitive to data fluctuation caused by numerical errors than traditional ROM. Since the tests of Basis CNNs still need fine-scale stiffness matrix and load vector, it can not be directly applied to nonlinear problems. The Coef CNNs can be applied to nonlinear problems and designed to determine the coefficients for linear combination of basis functions. In addition, two applications of CNN-based ROM are presented, including predicting MsFEM basis functions within oversampling regions and building accurate surrogates for inverse problems.

翻译：本文提出了一种结合卷积神经网络（CNN）与降阶建模（ROM）的高效多尺度问题模拟方法。此类问题通常由具有高维随机输入的偏微分方程描述。所提方法包含两个独立的CNN：基函数CNN（Basis CNNs）与系数CNN（Coef CNNs），分别对应ROM的两个核心组成部分，该方法被称为基于CNN的ROM。前者从细尺度解的快照中学习输入特定的基函数，并在输出层采用受Galerkin投影启发的激活函数，以从基函数重构细尺度解。数值结果表明，Basis CNNs学习到的基函数与数据特征高度吻合，从而显著减少了所需基函数的数量。此外，与传统ROM相比，基于CNN的ROM对数值误差引起的数据波动具有更强的鲁棒性。由于Basis CNNs的测试过程仍需细尺度刚度矩阵和载荷向量，该方法无法直接应用于非线性问题。而Coef CNNs可适用于非线性问题，其设计目标为确定基函数线性组合的系数。本文进一步展示了基于CNN的ROM的两个应用场景：在过采样区域内预测多尺度有限元法（MsFEM）基函数，以及为反问题构建精确的代理模型。