Learning macroscopic parameters in nonlinear multiscale simulations using nonlocal multicontinua upscaling techniques

Abstract

In this work, we present a novel nonlocal nonlinear coarse grid approximation using a machine learning algorithm. We consider unsaturated and two-phase flow problems in heterogeneous and fractured porous media, where mathematical models are formulated as general multicontinuum models. We construct a fine grid approximation using the finite volume method and embedded discrete fracture model. Macroscopic models for these complex nonlinear systems require nonlocal multicontinua approaches, which are developed in earlier works [8]. These rigorous techniques require complex local computations, which involve solving local problems in oversampled regions subject to constraints. The solutions of these local problems can be replaced by solving original problem on a coarse (oversampled) region for many input parameters (boundary and source terms) and computing effective properties derived by nonlinear nonlocal multicontinua approaches. The effective properties depend on many variables (oversampled region and the number of continua), thus their calculations require some type of machine learning techniques. In this paper, our contribution is two fold. First, we present macroscopic models and discuss how to effectively compute macroscopic parameters using deep learning algorithms. The proposed method can be regarded as local machine learning and complements our earlier approaches on global machine learning [39,38]. We consider a coarse grid approximation using two upscaling techniques with single phase upscaled transmissibilities and nonlocal nonlinear upscaled transmissibilities using a machine learning algorithm. We present results for two model problems in heterogeneous and fractured porous media and show that the presented method is highly accurate and provides fast coarse grid calculations.

Learning Adaptive Coarse Spaces of BDDC Algorithms for Stochastic Elliptic Problems with Oscillatory and High Contrast Coefficients

“…Here E[K](x) is the expected value of K(x, ω) and ξ i are independent and identically distributed Gaussian random variables with mean 0 and variance 1. As mentioned above, a few terms in KL expansion can give a reasonably accurate approximation, which can be explained now as we can just pick the dominant eigenfunctions in (11) and the other eigenvalues will decay rapidly. Therefore, KL expansion is an efficient method to represent the stochastic coefficient and favors the subsequent process.…”

“…From the KL expansion expression (11), we know the eigenvalues and eigenfunctions computed from the Fredholm integral equation are related to the covariance function of K(x, ω), which can be treated as the starting point of KL expansion. Throughout the experiments, two specially chosen covariance functions whose eigenvalues and eigenfunctions in the Fredholm integral equation can be computed analytically are considered with different expected values.…”

“…All these methods require a high computational power in realistic simulations and cannot be generalized to similar situations. In the past few decades, machine learning techniques have been applied on many aspects, such as image classification, fluid dynamics, and solving differential equations in [6][7][8][9][10][11][12][13][14]. In this paper, instead of directly solving the SPDE by machine learning, we choose a moderate step that involves the advantage of machine learning and, also, the accuracy from the adaptive BDDC algorithm.…”

Learning adaptive coarse spaces of BDDC algorithms for stochastic elliptic problems with oscillatory and high contrast coefficients

“…All these methods require a high computational power in realistic simulations and cannot be generalized to similar situations. In the past decades, machine learning technique has been applied on many aspects such as image classification, fluid dynamics and solving differential equations in [4,17,19,31,12,25,27,5,29]. In this paper, instead of directly solving the SPDE by machine learning, we choose a moderate step that involves the advantage of machine learning and also the accuracy from traditional numerical solver.…”

A deep learning based nonlinear upscaling method for transport equations

“…This fact motivates the use of deep learning. There are in the literature some works on using deep neural networks to learn macroscopic parameters in coarse scale or reduced order models; see for example [9,46,47,48,51]. The main idea is to consider the macroscopic variables as input and the downscaling functions or their average values as output.…”