Multiscale Model Reduction and Learning

Many engineering problems have multiscale features. These problems usually require some model reduction since the computational cost of a fine-scale solution is extremely expensive. Existing model reduction methods such as Generalized Multiscale Finite Element Method (GMsFEM) and Non-local multi-continuum approach (NLMC) have shown extensive success in solving multiscale problems especially on various flow simulation problems. However, there are still challenges in developing effective multiscale models for flow in more complicated heterogeneous media. The geometries of domain, coexistence of multiple continuum, and lack of observation data can all give rise to the difficulty of developing the reduced-order model. In this thesis, I will concentrate on the development of novel multiscale methods following the idea of the existing model reduction methods to address such problems. Moreover, deep learning techniques are combined to overcome certain difficulties met along model construction. These proposed models are targeted to tackle specific problems, where the performance is verified both numerically and analytically. For instance, flow simulation within a heterogeneous thin domain is one of such challenging problems. Though homogenization methods are proven to be successful when the media have clear scale separation, that’s not always the case for flow simulation within a capillary system. Using only one basis function in each coarse region can lead to large errors. We thus design a customized GMsFEM instead, which is able to automatically enrich the approximation space and significantly reduce the error. When simulating flow in a fractured vuggy reservoir, on the other hand, I develop a coarse solver under the framework of GMsFEM by combining it with multi-continuum model and Discrete Fracture Model (DFM). Instead of treating the media as a single continuum, I treat the multiscale formation hierarchically and consider it as a coupled system of matrix, fractures and vugs. This allows us to explicitly represent the mass transfers between continuum as well as model the local effects of the discrete fractures. We further investigate how deep learning can facilitate multiscale model construction for nonlinear flow dynamics. Utilizing a multi-layer neural network to approximate the reduced order model, the observed data can be easily incorporated to adjust the model. Deep learning techniques are also used to conduct model reduction. With a soft thresholding operator as an activation function, a novel neural network is proposed which can identify important multiscale features that are crucial in modeling the underlying flow. The forward input-output maps are thus learned in a reduced way. Extensive applications to engineering problems and numerical analysis are presented in supplement of the proposed approaches. It is shown that our proposed methods can significantly advance the computational efficiency and accuracy for multiscale flow simulation in various heterogeneous media