Related papers: Learning elliptic partial differential equations w…
Discovering hidden partial differential equations (PDEs) and operators from data is an important topic at the frontier between machine learning and numerical analysis. This doctoral thesis introduces theoretical results and deep learning…
Neural operators are a popular technique in scientific machine learning to learn a mathematical model of the behavior of unknown physical systems from data. Neural operators are especially useful to learn solution operators associated with…
Green's function characterizes a partial differential equation (PDE) and maps its solution in the entire domain as integrals. Finding the analytical form of Green's function is a non-trivial exercise, especially for a PDE defined on a…
We construct the first rigorously justified probabilistic algorithm for recovering the solution operator of a hyperbolic partial differential equation (PDE) in two variables from input-output training pairs. The primary challenge of…
PDE learning is an emerging field that combines physics and machine learning to recover unknown physical systems from experimental data. While deep learning models traditionally require copious amounts of training data, recent PDE learning…
Green's function provides an inherent connection between theoretical analysis and numerical methods for elliptic partial differential equations, and general absence of its closed-form expression necessitates surrogate modeling to guide the…
Many geometric and analytic properties of sets hinge on the properties of harmonic measure, notoriously missing for sets of higher co-dimension. The aim of this manuscript is to develop a version of elliptic theory, associated to a linear…
In this paper, we propose a data-driven framework for constructing efficient approximate inverse preconditioners for elliptic partial differential equations (PDEs) by learning the Green's function of the underlying operator with neural…
Learning the Green's function using deep learning models enables to solve different classes of partial differential equations. A practical limitation of using deep learning for the Green's function is the repeated computationally expensive…
We study estimates of the Green's function in $\mathbb{R}^d$ with $d \ge 2$, for the linear second order elliptic equation in divergence form with variable uniformly elliptic coefficients. In the case $d \ge 3$, we obtain estimates on the…
We propose a new data-driven approach for learning the fundamental solutions (Green's functions) of various linear partial differential equations (PDEs) given sample pairs of input-output functions. Building off the theory of functional…
We present two effective methods for solving high-dimensional partial differential equations (PDE) based on randomized neural networks. Motivated by the universal approximation property of this type of networks, both methods extend the…
Green's function plays a significant role in both theoretical analysis and numerical computing of partial differential equations (PDEs). However, in most cases, Green's function is difficult to compute. The troubles arise in the following…
Consider a discrete uniformly elliptic divergence form equation on the $d$ dimensional lattice $\Z^d$ with random coefficients. It has previously been shown that if the random environment is translational invariant, then the averaged…
Let $L$ be a second-order, homogeneous, constant (complex) coefficient elliptic system in ${\mathbb{R}}^n$. The goal of this article is provide a qualitative and quantitative study of the nature of the Green function associated with the…
We introduce a deep neural network based method for solving a class of elliptic partial differential equations. We approximate the solution of the PDE with a deep neural network which is trained under the guidance of a probabilistic…
Green's functions characterize the fundamental solutions of partial differential equations; they are essential for tasks ranging from shape analysis to physical simulation, yet they remain computationally prohibitive to evaluate on…
In this article we use linear algebra to improve the computational time for the obtaining of Green's functions of linear differential equations with reflection (DER). This is achieved by decomposing both the `reduced' equation (the ODE…
This work considers stochastic Galerkin approximations of linear elliptic partial differential equations (PDEs) with stochastic forcing terms and stochastic diffusion coefficients, that cannot be bounded uniformly away from zero and…
A linear algebraic method named the shifted conjugate-orthogonal-conjugate-gradient method is introduced for large-scale electronic structure calculation. The method gives an iterative solver algorithm of the Green's function and the…