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.

In this paper, we study the nonlinear stability of travelling wave solutions with shock profile for a relaxation model with a nonconvex flux, which is proposed by Jin and Xin [<i>Comm. Pure Appl. Math.</i>, 48 (1995), pp. 555--563] to approximate an original hyperbolic system numerically under the subcharacteristic condition introduced by T. P. Liu [<i>Comm. Math. Phys.</i>, 108 (1987), pp. 153--175]. The travelling wave solutions with strong shock profile are shown to be asymptotically stable under small disturbances with integral zero using an elementary but technical energy method. Proofs involve detailed study of the error equation for disturbances using the same weight function introduced in [<i>Comm. Math. Phys.</i>, 165 (1994), pp. 83--96].

Let Y be a non-singular projective manifold with an ample canonical sheaf, and let V be a Q-variation of Hodge structures of weight one on Y with Higgs bundle E1,0 ⊕ E0,1, coming from a family of Abelian varieties. If Y is a curve the Arakelov inequality says that the slopes satisfy μ(E1,0) − μ(E0,1) ≤ μ(­1 Y ). We prove a similar inequality in the higher dimensional case. If the latter is an equality, and if the discriminant of E1,0 or the one of E0,1 is zero, one hopes that Y is a Shimura variety, and V a uniformizing variation of Hodge structures. This is verified, in case the universal covering of Y does not contain factors of rank> 1. Part of the results extend to variations of Hodge structures over quasi-projective manifolds U.

We view regularized learning of a function in a Banach space from its finite samples as an optimization problem. Within the framework of reproducing kernel Banach spaces, we prove the representer theorem for the minimizer of regularized learning schemes with a general loss function and a nondecreasing regularizer. When the loss function and the regularizer are differentiable, a characterization equation for the minimizer is also established.

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