Deep Multiscale Model Learning

By Yating Wang, Siu Wun Cheung, Eric T. Chung, Yalchin R. Efendiev, Min Wang. 2018

84 citations recorded by the source corpus.

The objective of this paper is to design novel multi-layer neural network architectures for multiscale simulations of flows taking into account the observed data and physical modeling concepts. Our approaches use deep learning concepts combined with local multiscale model reduction methodologies to predict flow dynamics. Using reduced-order model concepts is important for constructing robust deep learning architectures since the reduced-order models provide fewer degrees of freedom. Flow dynamics can be thought of as multi-layer networks. More precisely, the solution (e.g., pressures and saturations) at the time instant $n+1$ depends on the solution at the time instant $n$ and input parameters, such as permeability fields, forcing terms, and initial conditions. One can regard the solution as a multi-layer network, where each layer, in general, is a nonlinear forward map and the number of layers relates to the internal time steps. We will rely on rigorous model reduction concepts to define unknowns and connections for each layer. In each layer, our reduced-order models will provide a forward map, which will be modified ("trained") using available data. It is critical to use reduced-order models for this purpose, which will identify the regions of influence and the appropriate number of variables. Because of the lack of available data, the training will be supplemented with computational data as needed and the interpolation between data-rich and data-deficient models. We will also use deep learning algorithms to train the elements of the reduced model discrete system. We will present main ingredients of our approach and numerical results. Numerical results show that using deep learning and multiscale models, we can improve the forward models, which are conditioned to the available data.

Subfields: Computational Mechanics, Computational Theory and Mathematics, Statistical and Nonlinear Physics

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Citations in the corpus

WEAK SINDY FOR PARTIAL DIFFERENTIAL EQUATIONS (2020, methodology): “In addition to the sparse regression approach adopted in SINDy, some of the primary techniques include Gaussian process regression [25], deep neural networks [27, 34, 17], Bayesian inference [44, 45, 37] and classical methods from numerical analysis [13, 15, 40].”
A Multiscale Neural Network Based on Hierarchical Matrices (2018, methodology): “used multiscale model reduction methods coupled with deep neural networks [47].”
Data Driven Governing Equations Approximation Using Deep Neural Networks (2018, methodology): “Eﬀorts have been devoted to the use of ANN for various aspects of scientiﬁc computing, including construction of reduced order model ([22]), aiding solution of conservation laws ([37]), multiscale problems ([8, 51]), solving and learning systems involving ODEs and PDEs 2.”