Related papers: Neural Green's Functions
The growing demand for accurate, efficient, and scalable solutions in computational mechanics highlights the need for advanced operator learning algorithms that can efficiently handle large datasets while providing reliable uncertainty…
Green's functions with continuum spectra are a way of avoiding the strong bounds on new physics from the absence of new narrow resonances in experimental data. We model such a situation with a five-dimensional model with two branes along…
This paper proposes a new way to learn Physics-Informed Neural Network loss functions using Generalized Additive Models. We apply our method by meta-learning parametric partial differential equations, PDEs, on Burger's and 2D Heat…
The two-dimensional elastodynamic Green tensor is the primary building block of solutions of linear elasticity problems dealing with nonuniformly moving rectilinear line sources, such as dislocations. Elastodynamic solutions for these…
On the unit square, we introduce a method for accurately computing source-neutral Green's functions of the fractional Laplacian operator with either periodic or homogeneous Neumann boundary conditions. This method involves analytically…
An important problem in many-body physics is to reconstruct the spectral density from the imaginary-time domain Green's function. Typically, the imaginary-time Green's function is generated by Monte Carlo methods. As the one-point fermionic…
Dyadic Green's function is an important tool of computational photonics, giving deeper insights into light-matter interaction. We present an operator approach to the derivation of the dyadic Green's function of a generic anisotropic…
Machine learning methods have been lately used to solve partial differential equations (PDEs) and dynamical systems. These approaches have been developed into a novel research field known as scientific machine learning in which techniques…
The recent rise of deep learning has led to numerous applications, including solving partial differential equations using Physics-Informed Neural Networks. This approach has proven highly effective in several academic cases. However, their…
Many physical processes such as weather phenomena or fluid mechanics are governed by partial differential equations (PDEs). Modelling such dynamical systems using Neural Networks is an active research field. However, current methods are…
Mixed-dimensional partial differential equations (PDEs) are characterized by coupled operators defined on domains of varying dimensions and pose significant computational challenges due to their inherent ill-conditioning. Moreover, the…
Simulations of complex physical systems are typically realized by discretizing partial differential equations (PDEs) on unstructured meshes. While neural networks have recently been explored for surrogate and reduced order modeling of PDE…
We provide an elementary derivation of the Green's function for Poisson's equation with Neumann boundary data on balls of arbitrary dimension, which was recently found in [Sadybekov et al., Eurasian Math. J. 7(2):100-105, 2016]. The…
Recently deep learning surrogates and neural operators have shown promise in solving partial differential equations (PDEs). However, they often require a large amount of training data and are limited to bounded domains. In this work, we…
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…
The aim of this paper is twofold. First, we prove $L^p$ estimates for a regularized Green's function in three dimensions. We then establish new estimates for the discrete Green's function and obtain some positivity results. In particular,…
In this paper we use a splitting technique to develop new multiscale basis functions for the multiscale finite element method (MsFEM). The multiscale basis functions are iteratively generated using a Green's kernel. The Green's kernel is…
Time-dependent partial differential equations (PDEs) for classic physical systems are established based on the conservation of mass, momentum, and energy, which are ubiquitous in scientific and engineering applications. These PDEs are…
Stieltjes boundary problems generalize the customary class of well-posed two-point boundary value problems in three independent directions, regarding the specification of the boundary conditions: (1) They allow more than two evaluation…
A Deep Learning approach is devised to estimate the elastic energy density $\rho$ at the free surface of an undulated stressed film. About 190000 arbitrary surface profiles h(x) are randomly generated by Perlin noise and paired with the…