Partial differential equations are used to model fluid flow in porous media. Neural networks can act as equation solution approximators by basing their forecasts on training samples of permeability maps and corresponding two-point flux approximation solutions. This paper illustrates how convolutional neural various architecture, depth parameter configurations manage forecast solutions the Darcy...

Abstract For an entire function solution of generalized bi-axisymmetric potential equation, we obtain a relationship between the growth characteristics and polynomial approximation errors in sup norm by using general functions introduced Seremeta [ On connection maximum modulus moduli coefficients its power series expansion , Amer. Math. Soc. Transl. 88 (1970), no. 2, 291–301].

A. Local and global Carleman estimates play a central role in the study of some partial differential equations regarding questions such as unique continuation and controllability. We survey and prove such estimates in the case of elliptic and parabolic operators by means of semi-classical microlocal techniques. Optimality results for these estimates and some of their consequences are pre...

In this paper we consider the following mathematical model: an elliptic boundary value problem, where the partial differential equation contains advection, diffusion, and deposition parts. A Monte Carlo (MC) method to solve this equation uses a local integral representation by the Green’s function and a random process called “Walks on Balls”(WOB). A new class of grid free MC algorithms for solv...

We present our recent results on the mathematical analysis of shock diffraction by two-dimensional convex cornered wedges in compressible fluid flow governed by the potential flow equation. The shock diffraction problem can be formulated as an initial-boundary value problem, which is invariant under the selfsimilar scaling. Then, by employing its self-similar invariance, the problem is reduced ...

In this article, an algorithm for the numerical approximation of two-phase flow in porous media by adaptive mesh is presented. A convergent and conservative finite volume scheme for an elliptic equation is proposed, together with the finite difference schemes, upwind and MUSCL, for a hyperbolic equation on grids with local refinement. Hence, an IMPES method is applied in an adaptive composite g...

Traditional numerical discretization-based solvers of partial differential equations (PDEs) are fundamentally agnostic to domains, boundary conditions and coefficients. In contrast, machine learnt have a limited generalizability across these elements value problems. This is strongly true in the case surrogate models that typically trained on direct simulations PDEs applied one specific problem....