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Efficient computation of graph diffusion equations (GDEs), such as Personalized PageRank, Katz centrality, and the Heat kernel, is crucial for clustering, training neural networks, and many other graph-related problems. Standard iterative…
The goal of this paper is to create a fruitful bridge between the numerical methods for approximating partial differential equations (PDEs) in fluid dynamics and the (iterative) numerical methods for dealing with the resulting large linear…
In a recent paper, Klaseboer et al. (IEEE Trans. Antennas Propag., vol. 65, no. 2, pp. 972-977, Feb. 2017) developed a surface integral formulation of electromagnetics that does not require working with integral equations that have singular…
We introduce the framework of continuous--depth graph neural networks (GNNs). Graph neural ordinary differential equations (GDEs) are formalized as the counterpart to GNNs where the input-output relationship is determined by a continuum of…
We prove generalized Gaffney inequalities and the discrete compactness for finite element differential forms on $s$-regular domains, including general Lipschitz domains. In computational electromagnetism, special cases of these results have…
Inverse problems involving systems of partial differential equations (PDEs) with many measurements or experiments can be very expensive to solve numerically. In a recent paper we examined dimensionality reduction methods, both stochastic…
In this work, we develop an efficient high order discontinuous Galerkin (DG) method for solving the Electrical Impedance Tomography (EIT). EIT is a highly nonlinear ill-posed inverse problem where the interior conductivity of an object is…
Generalized Chebyshev iteration (GCI) applied for solving linear equations with nonselfadjoint operators is considered. Sufficient conditions providing the convergence of iterations imposed on the domain of localization of the spectrum on…
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…
Partial differential equations (PDEs) are important tools to model physical systems and including them into machine learning models is an important way of incorporating physical knowledge. Given any system of linear PDEs with constant…
We consider a diffuse interface approach for solving an elliptic PDE on a given closed hypersurface. The method is based on a (bulk) finite element scheme employing numerical quadrature for the phase field function and hence is very easy to…
The recent development of spectral method has been praised for its high-order convergence in simulating complex physical problems. The combination of embedded boundary method and spectral method becomes a mainstream way to tackle…
In this work, we introduce new integral formulations based on the convolution quadrature method for the time-domain modeling of perfectly electrically conducting scatterers that overcome some of the most critical issues of the standard…
Implicit shape representation, such as SDFs, is a popular approach to recover the surface of a 3D shape as the level sets of a scalar field. Several methods approximate SDFs using machine learning strategies that exploit the knowledge that…
Recently, diffusion models have achieved great success in generative tasks. Sampling from diffusion models is equivalent to solving the reverse diffusion stochastic differential equations (SDEs) or the corresponding probability flow…
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…
The main objective of this paper is to develop a general method of geometric discretization for infinite-dimensional systems and apply this method to the EPDiff equation. The method described below extends one developed by Pavlov et al. for…
We propose a framework for solving partial differential equations (PDEs) motivated by isogeometric analysis (IGA) and local tensor-product splines. Instead of using a global basis for the solution space we use as generators the disjoint…
The free space Maxwell dielectric problem can be reduced to a system of surface integral equations (SIE). A numerical formulation for the Maxwell dielectric problem using an SIE system presents two key advantages: first, the radiation…
We propose a novel Skew Gradient Embedding (SGE) framework for systematically reformulating thermodynamically consistent partial differential equation (PDE) models-capturing both reversible and irreversible processes-as generalized gradient…