Related papers: Analytical Solving of Partial Differential Equatio…
Within recent years, considerable progress has been made regarding high-performance solvers for Partial Differential Equations (PDEs), yielding potential gains in efficiency compared to industry standard tools. However, the latter largely…
This paper investigates a numerical probabilistic method for the solution of some semilinear stochastic partial differential equations (SPDEs in short). The numerical scheme is based on discrete time approximation for solutions of systems…
We propose a new algorithm for solving parabolic partial differential equations (PDEs) and backward stochastic differential equations (BSDEs) in high dimension, by making an analogy between the BSDE and reinforcement learning with the…
The abstract mathematical theory of partial differential equations (PDEs) is formulated in terms of manifolds, scalar fields, tensors, and the like, but these algebraic structures are hardly recognizable in actual PDE solvers. The general…
High-dimensional partial differential equations (PDEs) pose significant challenges for numerical computation due to the curse of dimensionality, which limits the applicability of traditional mesh-based methods. Since 2017, the Deep BSDE…
Solving partial differential equations (PDEs) is a required step in the simulation of natural and engineering systems. The associated computational costs significantly increase when exploring various scenarios, such as changes in initial or…
This work presents a numerical analysis of computing transition states of semilinear elliptic partial differential equations (PDEs) via the index-1 saddle dynamics, or equivalently, the gentlest ascent dynamics. To establish clear…
Starting with some fundamental concepts, in this article we present the essential aspects of spectral methods and their applications to the numerical solution of Partial Differential Equations (PDEs). We start by using Lagrange and…
Solving systems of ordinary differential equations (ODEs) is essential when it comes to understanding the behavior of dynamical systems. Yet, automated solving remains challenging, in particular for nonlinear systems. Computer algebra…
We study a general class of singular degenerate parabolic stochastic partial differential equations (SPDEs) which include, in particular, the stochastic porous medium equations and the stochastic fast diffusion equation. We propose a fully…
In this paper, we propose some algorithms for analytical solution construction to nonlinear polynomial partial differential equations with constant function coefficients. These schemes are based on one-(single), two- (double) or three-…
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…
In recent years, data-driven methods have been developed to learn dynamical systems and partial differential equations (PDE). The goal of such work is discovering unknown physics and the corresponding equations. However, prior to achieving…
Neural operators have emerged as promising surrogate models for solving partial differential equations (PDEs), but struggle to generalise beyond training distributions and are often constrained to a fixed temporal discretisation. This work…
We propose a two-stage method called \textit{Spline Assisted Partial Differential Equation based Model Identification (SAPDEMI)} to identify partial differential equation (PDE)-based models from noisy data. In the first stage, we employ the…
In this paper we propose a new numerical method for solving stochastic differential equations (SDEs). As an application of this method we propose an explicit numerical scheme for a super linear SDE for which the usual Euler scheme diverges.
We propose a neural network-based meta-learning method to efficiently solve partial differential equation (PDE) problems. The proposed method is designed to meta-learn how to solve a wide variety of PDE problems, and uses the knowledge for…
We propose a collocation method based on multivariate polynomial splines over triangulation or tetrahedralization for the numerical solution of partial differential equations. We start with a detailed explanation of the method for the…
In this paper we explain how to use the Fast Fourier Transform (FFT) to solve partial differential equations (PDEs). We start by defining appropriate discrete domains in coordinate and frequency domains. Then describe the main limitation of…
In this paper we introduce and investigate a new kind of functional (including ordinary and evolutionary partial) differential equations. The main goal of this paper is to explore our new philosophy by some examples on functional ODEs and…