In this paper, we combine deep learning concepts and some proper orthogonal decomposition (POD) model reduction methods for predicting flow in heterogeneous porous media. Nonlinear flow dynamics is studied, where the dynamics is regarded as a multi-layer network. The solution at the current time step is regarded as a multi-layer network of the solution at the initial time and input parameters. As for input, we consider various sources, which include source terms (well rates), permeability fields, and initial conditions. We consider the flow dynamics, where the solution is known at some locations and the data is integrated to the flow dynamics by modifying the reduced-order model. This approach allows modifying the reduced-order formulation of the problem. Because of the small problem size, limited observed data can be handled. We consider enriching the observed data using the computational data in deep learning networks. The basis functions of the global reduced order model are selected such that the degrees of freedom represent the solution at observation points. This way, we can avoid learning basis functions, which can also be done using neural networks. We present numerical results, where we consider channelized permeability fields, where the network is constructed for various channel configurations. Our numerical results show that one can achieve a good approximation using forward feed maps based on multi-layer networks.

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In this paper, we develop and analyze a new multiscale discontinuous Galerkin (DG) method for one-dimensional stationary Schr\"{o}dinger equations with open boundary conditions \bo{which have highly oscillating solutions}. Our method uses a smaller finite element space than the WKB local DG (WKB-LDG) method while achieving the same order of accuracy with no resonance errors. We prove that the DG approximation converges optimally with respect to the mesh size $h$ in $L^2$ norm without the typical constraint \bo{that $h$ has to be smaller than the wave length}. Numerical experiments were carried out to verify the second order optimal convergence rate of the method and to demonstrate its ability to capture oscillating solutions on coarse meshes in the applications to Schr\"{o}dinger equations.

In this paper, we introduce a new adaptive data analysis method to study trend and instantaneous frequency of nonlinear and nonstationary data. This method is inspired by the Empirical Mode Decomposition method (EMD) and the recently developed compressed (compressive) sensing theory. The main idea is to look for the sparsest representation of multiscale data within the largest possible dictionary consisting of intrinsic mode functions of the form {a(t) cos(θ(t))}, where a ∈ V (θ), V (θ) consists of the functions smoother than cos(θ(t)) and θ   0. This problem can be formulated as a nonlinear l0 optimization problem. In order to solve this optimization problem, we propose a nonlinear matching pursuit method by generalizing the classical matching pursuit for the l0 optimization problem. One important advantage of this nonlinear matching pursuit method is it can be implemented very efficiently and is very stable to noise. Further, we provide an error analysis of our nonlinear matching pursuit method under certain scale separation assumptions. Extensive numerical examples will be given to demonstrate the robustness of our method and comparison will be made with the state-of-the-art methods. We also apply our method to study data without scale separation, and data with incomplete or under-sampled data.