Related papers: On the relations between two construction methods …
Learning the distance metric between pairs of samples has been studied for image retrieval and clustering. With the remarkable success of pair-based metric learning losses, recent works have proposed the use of generated synthetic points on…
We consider the equilibrium equations for a linearized Cosserat material and provide two perspectives concerning well-posedness. First, the system can be viewed as the Hodge Laplace problem on a differential complex. On the other hand, we…
For difference variational problems on lattice, this paper presents a relation between divergence variational symmetries and conservation laws for the associated Euler-Lagrange system provided by Noether's theorem. This hence inspires us to…
We use the general framework of summation-by-parts operators to construct conservative, energy-stable, and well-balanced semidiscretizations of two different nonlinear systems of dispersive shallow water equations with varying bathymetry:…
In this paper, we highlight the complimentary nature of the results of Anco & Bluman and Ibragimov in the construction of conservation laws; that whilst the former establishes the role of multipliers, the latter presents a formal procedure…
Energy preserving numerical methods for a certain class of PDEs are derived, applying the partition of unity method. The methods are extended to also be applicable in combination with moving mesh methods by the rezoning approach. These…
Kahan's method and a two-step generalization of the discrete gradient method are both linearly implicit methods that can preserve a modified energy for Hamiltonian systems with a cubic Hamiltonian. These methods are here investigated and…
In recent years, a significant amount of attention has been paid to solve partial differential equations (PDEs) by deep learning. For example, deep Galerkin method (DGM) uses the PDE residual in the least-squares sense as the loss function…
The fundamental idea of embedding a network in a metric space is rooted in the principle of proximity preservation. Nodes are mapped into points of the space with pairwise distance that reflects their proximity in the network. Popular…
In the classical Lagrangian approach to conservation laws of gauge-natural field theories a suitable (vector) density is known to generate the so--called {\em conserved Noether currents}. It turns out that along any section of the relevant…
In this dissertation we study basic local differential geometry, projective differential geometry, and prolongations of overdetermined geometric partial differential equations. It is simple to prolong an n-th order linear ordinary…
In this paper, a general theorem on the equivalence of pth moment stability between stochastic differential delay equations (SDDEs) and their numerical methods is proved under the assumptions that the numerical methods are strongly…
Reduced basis methods are popular for approximately solving large and complex systems of differential equations. However, conventional reduced basis methods do not generally preserve conservation laws and symmetries of the full order model.…
We present and analyze a new embedded--hybridized discontinuous Galerkin finite element method for the Stokes problem. The method has the attractive properties of full hybridized methods, namely an $H({\rm div})$-conforming velocity field,…
This paper presents identification of both network connected systems as well as distributed systems governed by PDEs in the framework of distributed optimization via the Alternating Direction Method of Multipliers. This approach opens first…
Learning distributed representations for nodes in graphs is a crucial primitive in network analysis with a wide spectrum of applications. Linear graph embedding methods learn such representations by optimizing the likelihood of both…
Symmetry properties of conservation laws of partial differential equations are developed by using the general method of conservation law multipliers. As main results, simple conditions are given for characterizing when a conservation law…
In Ref.~\cite{Sag} we proposed a geometric formulation of generalized Nambu mechanics. In the present paper we extend the class of Nambu systems by replacing the stringent condition of constancy of 3-form by closedness. We also explore the…
A discrete analogue of the dressing method is presented and used to derive integrable nonlinear evolution equations, including two infinite families of novel continuous and discrete coupled integrable systems of equations of nonlinear…
We introduce a new approach for designing numerical schemes for stochastic differential equations (SDEs). The approach, which we have called direction and norm decomposition method, proposes to approximate the required solution $X_t$ by…