Related papers: Learning Homogenization for Elliptic Operators
This study proposes a multi-field asymptotic homogenization for the analysis of thermo-piezoelectric materials with periodic microstructures. The effect of the microstructural heterogeneity is taken into account by means of periodic…
We introduce a new geometric approach for the homogenization and inverse homogenization of the divergence form elliptic operator with rough conductivity coefficients $\sigma(x)$ in dimension two. We show that conductivity coefficients are…
Numerical multiscale methods usually rely on some coupling between a macroscopic and a microscopic model. The macroscopic model is incomplete as effective quantities, such as the homogenized material coefficients or fluxes, are missing in…
Quantitative stochastic homogenization of linear elliptic operators is by now well-understood. In this contribution we move forward to the nonlinear setting of monotone operators with $p$-growth. This work is dedicated to a quantitative…
This article studies some numerical approximations of the homogenized matrix for stochastic linear elliptic partial differential equations in divergence form. We focus on the case when the underlying random field is a small perturbation of…
Learning the full family of solutions to parameterized partial differential equations (PDEs) is a central challenge to our ability to model the behavior of heterogeneous systems, with a variety of fundamental and application-oriented…
It is one of the most challenging problems in applied mathematics to approximatively solve high-dimensional partial differential equations (PDEs). Recently, several deep learning-based approximation algorithms for attacking this problem…
Recently, researchers have utilized neural networks to accurately solve partial differential equations (PDEs), enabling the mesh-free method for scientific computation. Unfortunately, the network performance drops when encountering a high…
We solve elliptic systems of equations posed on highly heterogeneous materials. Examples of this class of problems are composite structures and geological processes. We focus on a model problem which is a second-order elliptic equation with…
Partial differential equation (PDE) models with multiple temporal/spatial scales are prevalent in several disciplines such as physics, engineering, and many others. These models are of great practical importance but notoriously difficult to…
The neural network-based approach to solving partial differential equations has attracted considerable attention due to its simplicity and flexibility in representing the solution of the partial differential equation. In training a neural…
We approximate a diffusion equation with highly oscillatory coefficients with a diffusion equation with constant coefficients. The approach is put in action in contexts where only partial information (namely the global energy stored in the…
We propose a surrogate model for two-scale computational homogenization of elastostatics at finite strains. The macroscopic constitutive law is made numerically available via an explicit formulation of the associated macro-energy density.…
Numerical continuation calculations for ordinary differential equations (ODEs) are, by now, an established tool for bifurcation analysis in dynamical systems theory as well as across almost all natural and engineering sciences. Although…
Partial differential equations (PDEs) are central to computational electromagnetics (CEM) and photonic design, but classical solvers face high costs for large or complex structures. Quantum Hamiltonian simulation provides a framework to…
This work is a follow-up to our previous work "A numerical approach related to defect-type theories for some weakly random problems in homogenization" (preprint available on this archive). It extends and complements, both theoretically and…
Elastomeric mechanical metamaterials exhibit unconventional behaviour, emerging from their microstructures often deforming in a highly nonlinear and unstable manner. Such microstructural pattern transformations lead to non-local behaviour…
A computed approximation of the solution operator to a system of partial differential equations (PDEs) is needed in various areas of science and engineering. Neural operators have been shown to be quite effective at predicting these…
We propose a physics-informed consistency modeling framework for solving partial differential equations (PDEs) via fast, few-step generative inference. We identify a key stability challenge in physics-constrained consistency training, where…
There have been growing interests in leveraging experimental measurements to discover the underlying partial differential equations (PDEs) that govern complex physical phenomena. Although past research attempts have achieved great success…