Related papers: A New Numerical Method for Fast Solution of Partia…
We deal with the numerical solution of linear partial differential equations (PDEs) with focus on the goal-oriented error estimates including algebraic errors arising by an inaccurate solution of the corresponding algebraic systems. The…
Fully connected multilayer perceptrons are used for obtaining numerical solutions of partial differential equations in various dimensions. Independent variables are fed into the input layer, and the output is considered as solution's value.…
In this work, we present a hybrid numerical method for solving evolution partial differential equations (PDEs) by merging the time finite element method with deep neural networks. In contrast to the conventional deep learning-based…
We are interested in the numerical solution of nonsymmetric linear systems arising from the discretization of convection-diffusion partial differential equations with separable coefficients and dominant convection. Preconditioners based on…
In this article we introduce a simple straightforward and powerful method involving symbolic manipulation, Picard iteration, and auxiliary variables for approximating solutions of partial differential boundary value problems. The method is…
Recently, a nonlinear Poisson equation has been introduced to model nonlinear and nonlocal hyperpolarization effects in electrostatic solute-solvent interaction for biomolecular solvation analysis. Due to a strong nonlinearity associated…
Developing algorithms for solving high-dimensional partial differential equations (PDEs) has been an exceedingly difficult task for a long time, due to the notoriously difficult problem known as the "curse of dimensionality". This paper…
A novel method, connecting the space of solutions of a linear differential equation, of arbitrary order, to the space of monomials, is used for exploring the algebraic structure of the solution space. Apart from yielding new expressions for…
In recent years, tremendous progress has been made on numerical algorithms for solving partial differential equations (PDEs) in a very high dimension, using ideas from either nonlinear (multilevel) Monte Carlo or deep learning. They are…
This paper focuses on the numerical solution of elliptic partial differential equations (PDEs) with Dirichlet and mixed boundary conditions, specifically addressing the challenges arising from irregular domains. Both finite element method…
We propose a two-scale neural network method for solving partial differential equations (PDEs) with small parameters using physics-informed neural networks (PINNs). We directly incorporate the small parameters into the architecture of…
Traditional Monte Carlo integration using uniform random sampling exhibits degraded efficiency in low-regularity or high-dimensional problems. We propose a novel deep learning framework based on deterministic number-theoretic sampling…
The numerical methods for differential equation solution allow obtaining a discrete field that converges towards the solution if the method is applied to the correct problem. Nevertheless, the numerical methods have the restricted class of…
High-dimensional partial-differential equations (PDEs) arise in a number of fields of science and engineering, where they are used to describe the evolution of joint probability functions. Their examples include the Boltzmann and…
Self-consistent multi-particle simulation plays an important role in studying beam-beam effects and space charge effects in high-intensity beams. The Poisson equation has to be solved at each time-step based on the particle density…
A new fast multipole formulation for solving elliptic difference equations on unbounded domains and its parallel implementation are presented. These difference equations can arise directly in the description of physical systems, e.g.…
In the paper, we introduce a matrix method to constructively determine spaces of polynomial solutions (in general, multiplied by exponentials) to a system of constant coefficient linear PDE's with polynomial (multiplied by exponentials)…
This work develops a class of probabilistic algorithms for the numerical solution of nonlinear, time-dependent partial differential equations (PDEs). Current state-of-the-art PDE solvers treat the space- and time-dimensions separately,…
Computing solutions to partial differential equations using the fast Fourier transform can lead to unwanted oscillatory behavior. Due to the periodic nature of the discrete Fourier transform, waves that leave the computational domain on one…
In multi-phase fluid flow, fluid-structure interaction, and other applications, partial differential equations (PDEs) often arise with discontinuous coefficients and singular sources (e.g., Dirac delta functions). These complexities arise…