  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.
