Study models fractures in porous media using geometric analysis.
problem Analyzing fluid flow in fractures with complex geometries.
method Developed a geometric model using Riemannian manifold and Laplace Beltrami operators.
result Reduced model accurately approximates flow in complex fractures.
CNNs predict porosity, permeability, and tortuosity from porous media images.
problem Predicting key properties of porous media from images.
method Convolutional neural networks (CNNs) trained with lattice Boltzmann simulations.
result CNNs accurately predict porosity, permeability, and tortuosity.
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 g0 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)…
A new model reduces the cost of simulating fluid flow through porous materials.
problem High computational cost of simulating fluid flow through porous materials.
method Proposes a fully probabilistic, Darcy-type reduced-order model.
result The model significantly accelerates uncertainty quantification tasks.
Deep learning predicts fluid flow in porous media, accelerating simulations by orders of magnitude.
problem Accurate simulation of fluid flow in complex porous media requires excessive computational resources.
method Combining deep learning with direct simulation, using Gated U-Net CNNs trained on datasets of 2D and 3D porous media.
result Deep learning predictions can reach over 90% accuracy for permeability estimation and accelerate simulations by orders of magnitude.
Fractional porous media equations yield q-Gaussian solutions for stock price returns.
problem Modeling stock price returns using fractional porous media equations.
method Analyzed three types of fractional extensions of the porous media equation.
result Local and non-local fractional extensions fit S&P 500 data better than classical models.
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…
Bayesian networks link pore-scale to continuum-scale properties of porous media.
problem Understanding macroscopic properties from microscopic ones in porous media.
method Bayesian networks to model causal relationships and joint probability distributions.
result Causal relationships impact predictions of macroscopic properties from microscopic ones.
Develops a data-driven model for porous media flow simulations.
problem Capturing flow field and permeability in digital porous media.
method Data-driven approach using Lattice Boltzmann simulation data.
result Accurately predicts flow solutions with reduced computational time.
Stock market returns follow q-Gaussian distributions with super-diffusion.
problem Characterizing stock market price returns.
method Used q-Gaussian distributions and porous media equation to model stock market returns.
result Stock market returns follow q-Gaussian distributions with super-diffusion.
Deep Gaussian processes reduce uncertainty in porous media flow modeling.
problem Uncertainty quantification in flow through heterogeneous porous media.
method Multi-layer hierarchical Gaussian process with variational approximation.
result Automatic selection of hidden layer dimensions and uncertainty propagation.
In this paper, we first prove a localized Hamilton-type gradient estimate for the positive solutions of Porous Media type equations: ut=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.
problem Existence of complete Yamabe metrics with zero scalar curvature.
method Yamabe flow and local L1-stability analysis. result Local L1-stability of the Yamabe flow on manifolds with non-negative Ricci curvature. 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…
A new hybrid approach combines physics and machine learning for porous media transport.
problem Simulating 2-phase immiscible transport in porous media.
method Physics-informed deep learning with adversarial neural networks and automatic differentiation.
result The model accurately simulates shock and rarefaction phenomena with limited data.
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…
Deep learning framework for uncertainty quantification in physics.
problem Uncertainty in systems governed by non-linear differential equations.
method Physics-informed neural networks with adversarial inference.
result Effective training of deep generative models for physical systems.
The paper proves entropy formulae and Harnack estimates for porous medium equations on Riemannian manifolds.
problem Analyzing solutions to porous medium equations on Riemannian manifolds.
method Proves entropy formulae and differential Harnack estimates.
result Derives Harnack inequalities and Laplacian estimates as applications.
Paper extends Aronson-Bénilan estimates for porous medium equations on manifolds with negative curvature.
problem Estimating gradients for porous medium equations on manifolds with negative curvature.
method Develops Aronson-Bénilan gradient estimates for porous medium equations under lower bounds of N-weighted Ricci curvature with N<0. result Generalizes gradient estimates for porous medium equations to manifolds with negative curvature.
The paper establishes new gradient estimates for porous medium equations under Ricci flow.
problem Gradient estimates for porous medium equations under Ricci flow.
method Local Aronson-Benolan type gradient estimates for positive solutions of the porous medium equation under Ricci flow.
result Generalizes known gradient estimates to the Ricci flow context.
Paper derives gradient estimates for porous medium equations on Riemannian manifolds.
problem Gradient estimates for porous medium equations on Riemannian manifolds.
method Employing cutoff functions and the maximum principle.
result Derives Hamilton-Souplet-Zhang type gradient estimates for porous medium type equations.
The paper analyzes gradient estimates for a nonlinear heat equation on graphs.
problem Gradient estimates for the weighted porous medium equation on graphs.
method Analyzes the gradient estimates for the positive solutions of the weighted porous medium equation on graphs.
result Derives gradient estimates and Harnack inequality for the porous medium equation on graphs.
Researchers study fractional porous medium equation on hyperbolic space.
problem Analyzing the fractional porous medium equation on hyperbolic space.
method Existence results for solutions in weak sense, using fractional Laplacian and Green's function.
result Proves different smoothing effects for solutions.
Study nonnegative solutions on Riemannian manifolds using fractional porous medium equation.
problem Analyzing solutions to fractional porous medium equation on noncompact Riemannian manifolds.
method Existence and smoothing estimates for weak solutions in L1 and weighted spaces. result Results hold for Euclidean and hyperbolic spaces, including larger data classes.
Study improves Harnack estimates for porous medium equation under geometric flow.
problem Improving Harnack estimates for solutions to the porous medium equation under evolving metrics.
method Differential Harnack estimates for positive solutions to the porous medium equation with potential on time-dependent Riemannian metrics evolving by geometric flow.
result New Harnack estimates for the porous medium equation under geometric flow.
The study examines blow-up and global existence of solutions for porous medium equation on curved manifolds.
problem Analyzing blow-up and global existence of solutions for porous medium equation on negatively curved manifolds.
method Examined the porous medium equation with power-type reaction terms on negatively curved Riemannian manifolds.
result For p>m, small data give global solutions; for p<m, large data blow up in infinite time. This research proves that quadratic regularized optimal transport can approximate the Laplace-Beltrami operator on smooth manifolds.
problem Approximating the Laplace-Beltrami operator using optimal transport with quadratic regularization.
method Deriving first-order optimal potentials and analyzing the convergence of discrete Laplace operators.
result The discrete Laplace operators converge to the Laplace-Beltrami operator on smooth manifolds.
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.
problem Solving porous medium equation on noncompact manifolds with nonnegative Ricci curvature.
method Constructing a space X of functions larger than L1, in which the Green function on M appears as a weight, to solve the PME.
result The porous medium equation admits a solution in the weak dual sense for certain initial data.
Study tracks cryptocurrency news on social media.
problem Real-time tracking of relevant cryptocurrency news.
method Matched web news with social media tweets, analyzed tweet activity, and used machine learning models.
result Random forest autoregressive model performs well in predicting article mentions.
In this paper we study gradient estimates for the positive solutions of the porous medium equation: ut=Δum 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…
Study of long-time behavior of solutions on negatively curved manifolds.
problem Long-time behavior of solutions to the Porous Medium Equation on Cartan-Hadamard manifolds with negative curvature.
method Analysis of long-time behavior, proving existence and uniqueness of solutions, using comparison principles.
result Unexpected separate-variable behavior, reminiscent of Dirichlet problems on bounded Euclidean domains.
Toda flow explained as a porous medium equation.
problem Understanding the Toda flow through the lens of porous medium equations.
method Analyzing the geometry and dynamics of the porous medium equation and comparing it to the Toda flow.
result The Toda flow can be represented as a specific porous medium equation, revealing its gradient and Hamiltonian nature.
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…
We consider gradient estimates to positive solutions of porous medium equations and fast diffusion equations: ut=Δφ(up) 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…
SIP framework discovers governing equations in uncertain systems.
problem Discovering governing equations in systems with input variability and noisy data.
method SIP framework treats unknown coefficients as random variables and infers their posterior distribution by minimizing Kullback-Leibler divergence.
result SIP consistently identifies correct equations and lowers coefficient error by 82% relative to SINDy.
The paper proposes a method to infer user profiles from multiple sources of social media data.
problem Mining user profiles from social media data using a single type of information.
method Hinge-loss Markov Random Fields (HL-MRFs) integrated with multiple sources of UGC and social relations.
result HL-MRFs successfully incorporate multiple sources of information and outperform competing methods.
Study shows existence and uniqueness of solutions for porous medium equation on curved manifolds.
problem Existence and uniqueness of solutions for porous medium equation on curved manifolds.
method Very weak solutions, Cauchy problem, Cartan-Hadamard manifolds, Ricci curvature bounds, sectional curvature bounds.
result Sharp growth rate of initial data for existence, maximal existence time estimate, blow-up for specific manifolds and data.
Machine learning reduces DFN size by 80% for faster simulations.
problem Simulating flow and transport in large DFNs is computationally intensive.
method Graph theory and machine learning to identify a smaller, representative network.
result Reduced network size by approximately 20% without losing breakthrough curves.
The paper derives gradient estimates for porous medium and fast diffusion equations on metric measure spaces.
problem Gradient estimates for porous medium and fast diffusion equations on metric measure spaces.
method Derives Li-Yau and Souplet-Zhang type gradient estimates for the given equations.
result Gradient estimates for the equations on complete noncompact metric measure spaces with compact boundary.
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…
New method solves high-dimensional Bayesian inverse problems efficiently.
problem Efficiently solving high-dimensional Bayesian inverse problems with limited data.
method Physics-informed Neural Operators with RealNVP architecture for invertibility and differentiability.
result Accurate approximations of the full posterior without additional forward solves or sampling.
Deep learning predicts stock movements using social media data.
problem Predicting stock movements with traditional methods is challenging.
method Graph neural network combining financial data and social media.
result Improvement of 28% in cumulative returns.
Bayesian model reduces uncertainty in high-dimensional problems like random media.
problem Uncertainty in high-dimensional stochastic partial differential equations.
method Bayesian formulation for simultaneous dimension and model-order reduction.
result Sharp predictions with reduced model order and input dimensions.
Systematic review of ML models for detecting social media deception.
problem Detecting fake news, spam, and fake accounts on social media.
method 36 studies evaluated using PROBAST tool, identifying biases and limitations.
result Over-reliance on accuracy in imbalanced data settings is a flaw.