arXiv research

CNNs predict porosity, permeability, and tortuosity from porous media images.

In this paper, we study the discrete Morse flow for the Ricci flow on football, which is the 2-sphere with removed north and south poles and with the metric $g_0$ of constant scalar curvature, and and for Porous media equation on a bounded regular domain in the plane. We show that with a suitable assumption about $g(0)…

In this work, we analyze the flow filtration process of slightly compressible fluids in porous media containing man made fractures with complex geometries. We model the coupled fracture-porous media system where the linear Darcy flow is considered in porous media and the nonlinear Forchheimer equation is used inside th…

Fractional porous media equations yield q-Gaussian solutions for stock price returns.

Deep learning predicts fluid flow in porous media, accelerating simulations by orders of magnitude.

This work represents an application of constant mean curvature graphs (as solutions of the mean curvature PDE) to non-linear non-Darcy flows in porous media. It relates time invariant pressure distribution graphs to graphs of constant mean curvature surfaces. This differential geometric interpretation provides an impor…

Deep Gaussian processes reduce uncertainty in porous media flow modeling.

In this paper, we first prove a localized Hamilton-type gradient estimate for the positive solutions of Porous Media type equations: $u_t=ΔF(u),$ with $F'(u) > 0$, on a complete Riemannian manifold with Ricci curvature bounded from below. In the second part, we study Fast Diffusion Equation (FDE) and Porous Media Equ…

The paper studies Yamabe metrics and stability in Riemannian manifolds.

Microscopic (pore-scale) properties of porous media affect and often determine their macroscopic (continuum- or Darcy-scale) counterparts. Understanding the relationship between processes on these two scales is essential to both the derivation of macroscopic models of, e.g., transport phenomena in natural porous media,…

We construct exhaustion and cut-off functions with controlled gradient and Laplacian on manifolds with Ricci curvature bounded from below by a (possibly unbounded) nonpositive function of the distance from a fixed reference point, without any assumptions on the topology or the injectivity radius. Along the way we prove…

The objective for this work is to develop a data-driven proxy to high-fidelity numerical flow simulations using digital images. The proposed model can capture the flow field and permeability in a large verity of digital porous media based on solid grain geometry and pore size distribution by detailed analyses of the lo…

We analyze the Standard & Poor's 500 stock market index from the last 22 years. The probability density function of price returns exhibits two well-distinguished regimes with self-similar structure: the first one displays strong super-diffusion together with short-time correlations, and the second one corresponds to we…

We establish Schauder a priori estimates and regularity for solutions to a class of boundary-degenerate elliptic linear second-order partial differential equations. Furthermore, given a smooth source function, we prove regularity of solutions up to the portion of the boundary where the operator is degenerate. Degenerat…

In this paper, we consider three typical problems on a locally finite connected graph. The first one is to study the Bochner formula for the Laplacian operator on a locally finite connected graph. We use the Bochner formula to derive the Bernstein type estimate of the heat equation. The second is to derive the Reilly t…

Over the past few years, we developed a mathematically rigorous method to study the dynamical processes associated to nonlinear Forchheimer flows for slightly compressible fluids. We have proved the existence of a geometric transformation which relates constant mean curvature surfaces and time-invariant pressure distri…

Paper extends Aronson-Bénilan estimates for porous medium equations on manifolds with negative curvature.

Researchers study fractional porous medium equation on hyperbolic space.

Study nonnegative solutions on Riemannian manifolds using fractional porous medium equation.

In this paper, we study the gradient estimates for the positive solutions of the weighted porous medium equation $Δu^{m}=δ(x)u_{t}+ψu^{m}$ on graphs for $m>1$, which is a nonlinear version of the heat equation. Moreover, as applications, we derive a Harnack inequality and the estimates of the porous medium kernel on …

The present paper is devoted to the joint motion of two immiscible incompressible liquids in porous media. The liquids have different densities and initially separated by a surface of strong discontinuity (free boundary). We discuss the results of numerical simulations for exact free boundary problems on the microscopi…

Researchers solve porous medium equation on noncompact manifolds with Ricci curvature.

In this paper we study gradient estimates for the positive solutions of the porous medium equation: $u_t=Δu^m$ where $m>1$, which is a nonlinear version of the heat equation. We derive local gradient estimates of the Li-Yau type for positive solutions of porous medium equations on Riemannian manifolds with Ricci curv…

In this paper, we prove Perelman type $\mathcal{W}$-entropy formulae and global differential Harnack estimates for positive solutions to porous medium equation on the closed Riemannian manifolds with Ricci curvature bounded below. As applications, we derive Harnack inequalities and Laplacian estimates.

Direct numerical simulation of Stokes flow through an impermeable, rigid body matrix by finite elements requires meshes fine enough to resolve the pore-size scale and is thus a computationally expensive task. The cost is significantly amplified when randomness in the pore microstructure is present and therefore multipl…

In this paper we study the porous medium equation (PME) coupled with the Ricci flow on complete manifolds with bounded nonnegative curvature operator. In particular, we derive Aronson-Bénilan and Li-Yau-Hamilton type differential Harnack estimates for positive solutions to the PME, with a linear forcing term, under the…

Toda flow explained as a porous medium equation.

We consider gradient estimates to positive solutions of porous medium equations and fast diffusion equations: $u_t=Δ_φ(u^p)$ associated with the Witten Laplacian on Riemannian manifolds. Under the assumption that the $m$-dimensional Bakry-Emery Ricci curvature is bounded from below, we obtain gradient estimates which…

We present a deep learning framework for quantifying and propagating uncertainty in systems governed by non-linear differential equations using physics-informed neural networks. Specifically, we employ latent variable models to construct probabilistic representations for the system states, and put forth an adversarial …

We study the long-time behaviour of nonnegative solutions of the Porous Medium Equation posed on Cartan-Hadamard manifolds having very large negative curvature, more precisely when the sectional or Ricci curvatures diverge at infinity more than quadratically in terms of the geodesic distance to the pole. We find an une…

The paper derives gradient estimates for porous medium and fast diffusion equations on metric measure spaces.

We investigate existence and uniqueness of weak solutions of the Cauchy problem for the porous medium equation on negatively curved Riemannian manifolds. We show existence of solutions taking as initial condition a finite Radon measure, not necessarily positive. We then establish uniqueness in the class of nonnegative …

In this work we derive local gradient and Laplacian estimates of the Aronson-Bénilan and Li-Yau type for positive solutions of porous medium equations posed on Riemannian manifolds with a lower Ricci curvature bound. We also prove similar results for some fast diffusion equations. Inspired by Perelman's work we discove…

In this paper, we investigate some new local Aronson-Bénilan type gradient estimates for positive solutions of the porous medium equation $u_{t}=Δu^{m},$ under Ricci flow. As application, the related Harnack inequalities are derived. Our results generalize known results. These results in the paper can be regard as …

We consider the porous medium equation with power-type reaction terms $u^p$ on negatively curved Riemannian manifolds, and solutions corresponding to bounded, nonnegative and compactly supported data. If $p>m$, small data give rise to global-in-time solutions while solutions associated to large data blow up in finite t…

In this paper, by employ the cutoff function and the maximum principle, some Hamilton-Souplet-Zhang type gradient estimates for porous medium type equation are deduced. As a special case, an Hamilton-Souplet-Zhang type gradient estimates of the heat equation is derived which is different from the result of Souplet-Zhan…

New method solves high-dimensional Bayesian inverse problems efficiently.

The paper solves a complex financial optimization problem using a novel mathematical technique.

Study shows conditions for nonexistence of solutions in Riemannian geometry.

Let $(M, g(t))$, $t\in[0,T)$ be a closed Riemannian $n$-manifold whose Riemannian metric $g(t)$ evolves by the geometric flow $\frac{\partial }{\partial t} g_{ij}=-2S_{ij}$, where $S_{ij}(t)$ is a symmetric two-tensor on $(M,g(t))$. We discuss differential Harnack estimates for positive solution to the porous medium …

A new training method improves stability and generalization of DeepONets.

Data assimilation for parameter and state estimation in subsurface transport problems remains a significant challenge due to the sparsity of measurements, the heterogeneity of porous media, and the high computational cost of forward numerical models. We present a physics-informed deep neural networks (DNNs) machine lea…

New equations for Cosserat media motions derived from bundle automorphisms.

Global solutions and smoothing effects for reaction-diffusion equations on manifolds.

Structural and topological information play a key role in modeling flow and transport through fractured rock in the subsurface. Discrete fracture network (DFN) computational suites such as dfnWorks are designed to simulate flow and transport in such porous media. Flow and transport calculations reveal that a small back…

This article asks how planning scholarship may effectively gain impact in planning practice through media exposure. In liberal democracies the public sphere is dominated by mass media. Therefore, working with such media is a prerequisite for effective public impact of planning research. Using the example of megaproject…

Study compares sentiment spillover networks from news and social media in tech companies.

As a highlighting research topic in the multimedia area, cross-media retrieval aims to capture the complex correlations among multiple media types. Learning better shared representation and distance metric for multimedia data is important to boost the cross-media retrieval. Motivated by the strong ability of deep neura…