相关论文: Errors Theory using Dirichlet Forms, Linear Partia…
We present the application of the variational-wavelet analysis to the analysis of quantum ensembles in Wigner framework. (Naive) deformation quantization, the multiresolution representations and the variational approach are the key points.…
The discovery of partial differential equations (PDEs) is a challenging task that involves both theoretical and empirical methods. Machine learning approaches have been developed and used to solve this problem; however, it is important to…
A core challenge in Machine Learning is to learn to disentangle natural factors of variation in data (e.g. object shape vs. pose). A popular approach to disentanglement consists in learning to map each of these factors to distinct subspaces…
A physics-informed neural network is developed to solve conductive heat transfer partial differential equation (PDE), along with convective heat transfer PDEs as boundary conditions (BCs), in manufacturing and engineering applications where…
The estimation of the probability of rare events is an important task in reliability and risk assessment. We consider failure events that are expressed in terms of a limit-state function, which depends on the solution of a partial…
In this thesis we investigate how the nonlocalities affect the study of different PDEs coming from physics, and we analyze these equations under almost optimal assumptions of the nonlinearity. In particular, we focus on the fractional…
This article presents a general approach akin to domain-decomposition methods to solve a single linear PDE, but where each subdomain of a partitioned domain is associated to a distinct variational formulation coming from a mutually…
In this paper we investigate the homogenization problem with a non-homogeneous Dirichlet condition. Our aim is to give error estimates with boundary data in $H^{1/2}(\partial\Omega)$. The tools used are those of the unfolding method in…
This paper is concerned with approximations and related discretization error estimates for the normal derivatives of solutions of linear elliptic partial differential equations. In order to illustrate the ideas, we consider the Poisson…
Recent works have shown that deep neural networks can be employed to solve partial differential equations, giving rise to the framework of physics informed neural networks. We introduce a generalization for these methods that manifests as a…
Solving parametric Partial Differential Equations (PDEs) for a broad range of parameters is a critical challenge in scientific computing. To this end, neural operators, which \textcolor{black}{predicts the PDE solution with variable PDE…
We establish error estimates for the approximation of parametric $p$-Dirichlet problems deploying the Deep Ritz Method. Parametric dependencies include, e.g., varying geometries and exponents $p\in (1,\infty)$. Combining the derived error…
An integral representation of solutions of the wave equation as a superposition of other solutions of this equation is built. The solutions from a wide class can be used as building blocks for the representation. Considerations are based on…
Wavelet based algorithms in numerical analysis are similar to other transform methods in that vectors and operators are expanded into a basis and the computations take place in this new system of coordinates. However, due to the recursive…
The analysis of wave propagation problems in linear damped media must take into account both propagation features and attenuation process. To perform accurate numerical investigations by the finite differences or finite element method, one…
This paper is a continuation of [arXiv:1603.02204]. Exploded layered tropical (ELT) algebra is an extension of tropical algebra with a structure of layers. These layers allow us to use classical algebraic results in order to easily prove…
Minimizing the so-called "Dirichlet energy" with respect to the domain under a volume constraint is a standard problem in shape optimization which is now well understood. This article is devoted to a prototypal non-linear version of the…
The inversion theorem and convolution theorem of the conformable fractional Laplace transforms are developed. All the elementary properties of the classical Laplace transform are extended to the conformable fractional transform, and using…
Employing a limiting case of a conjecture for constructing piecewise separable-variables functions, the elements of the Pseudoanalytic Function Theory are used for numerically approaching solutions of the forward Dirichlet boundary value…
The Convex Envelope of a given function was recently characterized as the solution of a fully nonlinear Partial Differential Equation (PDE). In this article we study a modified problem: the Dirichlet problem for the underlying PDE. The main…