Related papers: Deep Generalized Green's Functions
In this work we apply the Deep Galerkin Method (DGM) described in Sirignano and Spiliopoulos (2018) to solve a number of partial differential equations that arise in quantitative finance applications including option pricing, optimal…
Deep neural networks are powerful tools for approximating functions, and they are applied to successfully solve various problems in many fields. In this paper, we propose a neural network-based numerical method to solve partial differential…
Partial Differential Equations (PDE) are fundamental to model different phenomena in science and engineering mathematically. Solving them is a crucial step towards a precise knowledge of the behaviour of natural and engineered systems. In…
GreenLearning networks (GL) directly learn Green's function in physical space, making them an interpretable model for capturing unknown solution operators of partial differential equations (PDEs). For many PDEs, the corresponding Green's…
This paper presents a novel {\em Interpolated Factored Green Function} method (IFGF) for the accelerated evaluation of the integral operators in scattering theory and other areas. Like existing acceleration methods in these fields, the IFGF…
Spherical deep learning has been widely applied to a broad range of real-world problems. Existing approaches often face challenges in balancing strong spherical geometric inductive biases with the need to model real-world heterogeneity. To…
Option pricing often requires solving partial differential equations (PDEs). Although deep learning-based PDE solvers have recently emerged as quick solutions to this problem, their empirical and quantitative accuracy remain not well…
This paper introduces a new Windowed Green Function (WGF) method for the numerical integral-equation solution of problems of electromagnetic scattering by obstacles in presence of dielectric or conducting half-planes. The WGF method, which…
Numerically solving high-dimensional partial differential equations (PDEs) is a major challenge. Conventional methods, such as finite difference methods, are unable to solve high-dimensional PDEs due to the curse-of-dimensionality. A…
General formula for causal Green's function of linear differential operator of given degree in one variable is given according to coefficient functions of differential operator as a series of integrals. The solution also provides analytic…
We propose a mesh refinement technique for solving elliptic difference equations on unbounded domains based on the fast lattice Green's function (FLGF) method. The FLGF method exploits the regularity of the Cartesian mesh and uses the fast…
We present the variational multiscale (VMS) method for partial differential equations (PDEs) with stochastic coefficients and source terms. We use it as a method for generating accurate coarse-scale solutions while accounting for the effect…
The single-particle Green's function (GF) of mesoscopic structures plays a central role in mesoscopic quantum transport. The recursive GF technique is a standard tool to compute this quantity numerically, but it lacks physical transparency…
The finite element method, finite difference method, finite volume method and spectral method have achieved great success in solving partial differential equations. However, the high accuracy of traditional numerical methods is at the cost…
We consider some non-linear non-homogeneous partial differential equations (PDEs) and derive their exact Green function solution as a functional Taylor expansion in powers of the source. The kind of PDEs we consider are dispersive ones…
The aim of this article is to provide a firm mathematical foundation for the application of deep gradient flow methods (DGFMs) for the solution of (high-dimensional) partial differential equations (PDEs). We decompose the generalization…
High-dimensional PDEs have been a longstanding computational challenge. We propose to solve high-dimensional PDEs by approximating the solution with a deep neural network which is trained to satisfy the differential operator, initial…
This paper presents a windowed Green function (WGF) method for the numerical solution of problems of elastic scattering by "locally-rough surfaces" (i.e., local perturbations of a half space), under either Dirichlet or Neumann boundary…
Linear partial differential equations (PDEs) are an important, widely applied class of mechanistic models, describing physical processes such as heat transfer, electromagnetism, and wave propagation. In practice, specialized numerical…
Recent works have shown that deep neural networks can be employed to solve partial differential equations, giving rise to the framework of physics informed neural networks. We introduce a generalization for these methods that manifests as a…