Related papers: Learning Homogenization for Elliptic Operators
We consider the large-scale regularity of solutions to second-order linear elliptic equations with random coefficient fields. In contrast to previous works on regularity theory for random elliptic operators, our interest is in the…
Computational multiscale methods for analyzing and deriving constitutive responses have been used as a tool in engineering problems because of their ability to combine information at different length scales. However, their application in a…
Identifying parameters in partial differential equations (PDEs) represents a very broad class of applied inverse problems. In recent years, several unsupervised learning approaches using (deep) neural networks have been developed to solve…
The operator learning has received significant attention in recent years, with the aim of learning a mapping between function spaces. Prior works have proposed deep neural networks (DNNs) for learning such a mapping, enabling the learning…
We study the finite element approximation of linear second-order elliptic partial differential equations in nondivergence form with highly heterogeneous diffusion and drift coefficients. A generalized Cordes condition is imposed to…
The structural properties of mechanical metamaterials are typically studied with two-scale methods based on computational homogenization. Because such materials have a complex microstructure, enriched schemes such as second-order…
We introduce a new concept of sparsity for the stochastic elliptic operator $-{\rm div}\left(a(x,\omega)\nabla(\cdot)\right)$, which reflects the compactness of its inverse operator in the stochastic direction and allows for spatially…
A major challenge in flow through porous media is to better understand the link between microstructure and macroscale flow and transport. For idealised microstructures, the mathematical framework of homogenisation theory can be used for…
Embeddings provide low-dimensional representations that organize complex function spaces and support generalization. They provide a geometric representation that supports efficient retrieval, comparison, and generalization. In this work we…
Developing efficient numerical algorithms for the solution of high dimensional random Partial Differential Equations (PDEs) has been a challenging task due to the well-known curse of dimensionality. We present a new solution framework for…
Dimensionless learning is a data-driven framework for discovering dimensionless numbers and scaling laws from experimental measurements. This tutorial introduces the method, explaining how it transforms experimental data into compact…
In this work, we develop a method for learning interpretable, thermodynamically stable and Galilean invariant partial differential equations (PDEs) based on the Conservation-dissipation Formalism of irreversible thermodynamics. As governing…
We study the averaging behavior of nonlinear uniformly elliptic partial differential equations with random Dirichlet or Neumann boundary data oscillating on a small scale. Under conditions on the operator, the data and the random media…
Pattern formation is a widely observed phenomenon in diverse fields including materials physics, developmental biology and ecology, among many others. The physics underlying the patterns is specific to the mechanisms, and is encoded by…
This work focuses on computing the homogenized elastic properties of rocks from 3D micro-computed-tomography (micro-CT) scanned images. The accurate computation of homogenized properties of rocks, archetypal random media, requires both…
We consider the numerical reconstruction of the spatially dependent conductivity coefficient and the source term in elliptic partial differential equations in a two-dimensional convex polygonal domain, with the homogeneous Dirichlet…
In many complex applications, data heterogeneity and homogeneity exist simultaneously. Ignoring either one will result in incorrect statistical inference. In addition, coping with complex data that are non-Euclidean becomes more common. To…
This thesis is divided into five chapters. The aim is the study of the effectiveness of a chemical as defined by R. Aris for semilinear elliptic equations. The first chapter focuses on homogenization on quasilinear diffusion-reaction…
We introduce a new constructive method for establishing lower bounds on convergence rates of periodic homogenization problems associated with divergence type elliptic operators. The construction is applied in two settings. First, we show…
In recent years, data-driven methods have been developed to learn dynamical systems and partial differential equations (PDE). The goal of such work is discovering unknown physics and the corresponding equations. However, prior to achieving…