相关论文: Analytical Solving of Partial Differential Equatio…

Partial differential equations (PDEs) are ubiquitous in the world around us, modelling phenomena from heat and sound to quantum systems. Recent advances in deep learning have resulted in the development of powerful neural solvers; however,…

In this paper we discuss three symbolic approaches for the generation of a finite difference scheme of a partial differential equation (PDE). We prove, that for a linear PDE with constant coefficients these three approaches are equivalent…

We describe a neural-based method for generating exact or approximate solutions to differential equations in the form of mathematical expressions. Unlike other neural methods, our system returns symbolic expressions that can be interpreted…

Algorithms are presented for the tanh- and sech-methods, which lead to closed-form solutions of nonlinear ordinary and partial differential equations (ODEs and PDEs). New algorithms are given to find exact polynomial solutions of ODEs and…

The quest for analytical solutions to differential equations has traditionally been constrained by the need for extensive mathematical expertise. Machine learning methods like genetic algorithms have shown promise in this domain, but are…

Motivated by the remarkable success of artificial intelligence (AI) across diverse fields, the application of AI to solve scientific problems, often formulated as partial differential equations (PDEs), has garnered increasing attention.…

This paper develops a probabilistic numerical method for solution of partial differential equations (PDEs) and studies application of that method to PDE-constrained inverse problems. This approach enables the solution of challenging inverse…

We are concerned with the arithmetic of solutions to ordinary or partial nonlinear differential equations which are algebraic in the indeterminates and their derivatives. We call these solutions D-algebraic functions, and their equations…

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…

This paper presents a generalised symbolic algorithm for solving systems of linear algebraic equations with multi-diagonal coefficient matrices. The algorithm is given in a pseudocode. A theorem which gives the condition for correctness of…

Partial Differential Equations (PDEs) have long been recognized as powerful tools for image processing and analysis, providing a framework to model and exploit structural and geometric properties inherent in visual data. Over the years,…

Partial differential equation (PDE) is an important math tool in science and engineering. This paper experimentally demonstrates an optical neural PDE solver by leveraging the back-propagation-free on-photonic-chip training of…

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,…

There are many numerical methods for solving partial different equations (PDEs) on manifolds such as classical implicit, finite difference, finite element, and isogeometric analysis methods which aim at improving the interoperability…

This book encompasses both traditional and modern methods treating partial differential equation (PDE) of first order and second order. There is a balance in making a selfcontained mathematical text and introducing new subjects. The Lie…

The numerical solution methods for partial differential equation (PDE) solution allow obtaining a discrete field that converges towards the solution if the method is applied to the correct problem. Nevertheless, the numerical methods…

A template-based generic programming approach was presented in a previous paper that separates the development effort of programming a physical model from that of computing additional quantities, such as derivatives, needed for embedded…

The numerical solution of differential equations can be formulated as an inference problem to which formal statistical approaches can be applied. However, nonlinear partial differential equations (PDEs) pose substantial challenges from an…

The paper develops the method for construction of families of particular solutions to some classes of nonlinear Partial Differential Equations (PDE). Method is based on the specific link between algebraic matrix equations and PDE.…

This paper introduces PDEformer, a neural solver for partial differential equations (PDEs) capable of simultaneously addressing various types of PDEs. We propose to represent the PDE in the form of a computational graph, facilitating the…