偏微分方程(partial differential equations,简称PDE)是数学中的一个重要分支,研究方程中包含多个未知函数及其偏导数的关系。本文将介绍偏微分方程的基本概念和解法。 二、基本概念 2.1 偏导数 偏导数是函数在某一点上对某个自变量的变化率。对于二元函数 ,其偏导数可表示为 和 。类似地,对于三元函数 ,其偏导数可表...
《Lectures on Partial Differential Equations》以及《Mathematical Aspects of Classical and Celestial Mecha...
Physics-informed neural networks (PINNs) have emerged as a fundamental approach within deep learning for the resolution of partial differential equations (PDEs). Nevertheless, conventional multilayer perceptrons (MLPs) are characterized by a lack of interpretability and encounter the spectral bias problem,...
According to the theorem, we can express this as a combination of single-variable functions. This allows us to break down the multivariable equation into several individual equations, each involving one variable and another function of it, and so forth. Then sum up the outputs of all of...
He has made fundamental contributions in dynamical systems, singularity theory, stability theory, topology, algebraic geometry, magneto-hydrodynamics, partial differential equations, and other areas. Professor Arnol′d has won numerous honors and awards, including the Lenin Prize, the Crafoord Prize, and...
Differential Equations》,《常微分方程的几何方法》, 《Lectures on Partial Differential Equations》以及...
(financed by military and other institutions dealing with missiles, such as NASA.).Cryptography has generated number theory, algebraic geometry over finite fields, algebra, combinatorics and computers.Hydrodynamics recreated complex analysis, partial differential equations, Lie groups and algebra theory, ...
In the seminal study, DeepONet effectively mapped between unknown parametric functions and solution spaces for several linear and nonlinear partial differential equations (PDEs) while also learning explicit operators such as integrals. DeepONets have been increasingly used to tackle scientific and ...
AI for partial differential equations (PDEs) has garnered significant attention, particularly with the emergence of Physics-informed neural networks (PINNs). The recent advent of Kolmogorov-Arnold Network (KAN) indicates that there is potential to revisit and enhance the previously MLP-based PINNs. ...
We explore both B-spline and wavelet-based implementations of PIKAN and benchmark their performance across various ordinary and partial differential equations using unsupervised (data-free) and supervised (data-driven) techniques. For certain differential equations, the data-free approach suffices to ...