Convolutional neural network based reduced order modeling for multiscale problems

TL;DR

This paper introduces a CNN-based reduced order modeling approach for multiscale problems, combining basis and coefficient CNNs to efficiently approximate solutions of PDEs with high-dimensional random inputs.

Contribution

The novel CNN-based ROM method learns basis functions and coefficients, reducing computational complexity and improving robustness over traditional ROM in multiscale PDE problems.

Findings

Basis CNNs produce basis functions resembling data, reducing basis number.

CNN-based ROM is less sensitive to data fluctuation and numerical errors.

Method successfully predicts basis functions and builds surrogates for inverse problems.

Abstract

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.