Related papers: New multidimensional partially integrable generali…
Complete discrimination system for polynomial and direct integral method were discussed systematically. In particularly, we pointed out some mistaken viewpoints. Combining with complete discrimination system for polynomial, direct integral…
Recent work has introduced a simple numerical method for solving partial differential equations (PDEs) with deep neural networks (DNNs). This paper reviews and extends the method while applying it to analyze one of the most fundamental…
Developing efficient numerical algorithms for the solution of high dimensional random Partial Differential Equations (PDEs) has been a challenging task due to the well-known curse of dimensionality. We present a new solution framework for…
A nonconformal domain decomposition method based on the hybrid surface integral equation partial differential equation (SIE-PDE) formulation is proposed to solve the transverse magnetic electromagnetic problems. In the hybrid SIE-PDE…
Partial differential equations (PDEs) with spatially-varying coefficients arise throughout science and engineering, modeling rich heterogeneous material behavior. Yet conventional PDE solvers struggle with the immense complexity found in…
New extensions of the KP and modified KP hierarchies with self-consistent sources are proposed. The latter provide new generalizations of $(2+1)$-dimensional integrable equations, including the DS-III equation and the $N$-wave problem.…
One way of constructing explicit expressions of solutions of integrable systems of Partial Differential Equations (PDEs) goes via the Darboux method. This requires the construction of Darboux matrices. Here we introduce a novel algorithm to…
In the article we discuss the notion of the generalized invariant manifold introduced in our previous study. In the literature the method of the differential constraints is well known as a tool for constructing particular solutions for the…
Numerical solution of partial differential equations (PDEs) plays a vital role in various fields of science and engineering. In recent years, deep neural networks (DNNs) have emerged as a powerful tool for solving PDEs, leveraging their…
Parametric partial differential equations (PDEs) are fundamental for modeling a wide range of physical and engineering systems influenced by uncertain or varying parameters. Traditional neural network-based solvers, such as Physics-Informed…
Wave propagation problems are typically formulated as partial differential equations (PDEs) on unbounded domains to be solved. The classical approach to solving such problems involves truncating them to problems on bounded domains by…
Physics-informed machine learning (PIML) has emerged as a promising alternative to conventional numerical methods for solving partial differential equations (PDEs). PIML models are increasingly built via deep neural networks (NNs) whose…
Remarkable parallelism between the theory of integrable systems of first-order quasilinear PDE and some old results in projective and affine differential geometry of conjugate nets, Laplace equations, their Bianchi-Baecklund transformations…
We propose novel less diffusive schemes for conservative one- and two-dimensional hyperbolic systems of nonlinear partial differential equations (PDEs). The main challenges in the development of accurate and robust numerical methods for the…
A hybrid surface integral equation partial differential equation (SIE-PDE) formulation without the boundary condition requirement is proposed to solve the transverse magnetic (TM) electromagnetic problems. In the proposed formulation, the…
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…
Realistic physical phenomena exhibit random fluctuations across many scales in the input and output processes. Models of these phenomena require stochastic PDEs. For three-dimensional coupled (vector-valued) stochastic PDEs (SPDEs), for…
The main object of the paper is a recently discovered family of multicomponent integrable systems of partial differential equations, whose particular cases include many well-known equations such as the Korteweg--de Vries, coupled KdV, Harry…
In [Alekseevsky, Gutt, Manno, Moreno: "A general method to construct invariant PDEs on homogeneous manifolds", Communications in Contemporary Mathematics (2021)] the authors have developed a method for constructing $G$-invariant PDEs…
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…