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The numerical simulation of fluid flow through a complex geometry with heat transfer is of strong interest for many applications, such as oil-filled power transformers. A fundamental challenge here is that high resolution is necessary to…
Trefftz schemes are high-order Galerkin methods whose discrete spaces are made of elementwise exact solutions of the underlying PDE. Trefftz basis functions can be easily computed for many PDEs that are linear, homogeneous, and have…
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…
Numerical analytical heat transfer models play a critical role in geothermal design and feasibility studies. Classical solutions, such as those proposed by Gringarten et al. 1975, rely on simplified assumptions and systematically…
Computation of electromagnetic fields due to point sources (Hertzian dipoles) in cylindrically stratified media is a classical problem for which analytical expressions of the associated tensor Green's function have been long known. However,…
In this paper, we derive closed form solutions for the quasi-stationary problems of moving heat sources within the gradient theory of heat transfer. This theory can be formally deduced from the two-temperature model and it can be treated as…
We consider a diffusion process on an evolving surface with a piecewise Lipschitz-continuous boundary from an energetic point of view. We employ an energetic variational approach with both surface divergence and transport theorems to derive…
The aim of this paper is to provide a mathematical analysis of transformer architectures using a self-attention mechanism with layer normalization. In particular, observed patterns in such architectures resembling either clusters or uniform…
Generalized Hydrodynamics (GHD) has recently been devised as a method to solve the dynamics of integrable quantum many-body systems beyond the mean-field approximation. In its original form, a major limitation is the inability to predict…
This paper exploits the theory of geometric gradient flows to introduce an alternative regularization of the thin-film equation. The solution properties of this regularization are investigated via a sequence of numerical simulations whose…
This article presents a theoretical analysis of a one-dimensional heat transfer problem in two layers involving diffusion, advection, internal heat generation or loss linearly dependent on temperature in each layer, and heat generation due…
Convection-diffusion equations provide the basis for describing heat and mass transfer phenomena as well as processes of continuum mechanics. To handle flows in porous media, the fundamental issue is to model correctly the convective…
"Generalized Hydrodynamics" (GHD) stands for a model that describes one-dimensional \textit{integrable} systems in quantum physics, such as ultra-cold atoms or spin chains. Mathematically, GHD corresponds to nonlinear equations of kinetic…
Graph convolution is the core of most Graph Neural Networks (GNNs) and usually approximated by message passing between direct (one-hop) neighbors. In this work, we remove the restriction of using only the direct neighbors by introducing a…
The analogy to heat diffusion has enhanced our understanding of information flow in graphs and inspired the development of Graph Neural Networks (GNNs). However, most diffusion-based GNNs emulate passive heat diffusion, which still suffers…
When the variations of surface temperature are measured both spatially and temporally, analytical expressions that correctly account for multi-dimensional transient conduction can be applied. To enhance the accessibility of these accurate…
Continuous graph neural models based on differential equations have expanded the architecture of graph neural networks (GNNs). Due to the connection between graph diffusion and message passing, diffusion-based models have been widely…
A construction of a pseudo-differential operator on non-archimedean local fields invariant under a finite group action is given together with the solution of the corresponding Cauchy problem. This construction is applied to parts of the…
Normal and anomalous diffusion are ubiquitous in many complex systems [1] . Here, we define a time and space generalized diffusion equation (GDE), which uses fractional-time derivatives and transformed d-path Laplacian operators on…
We study evolution equations on metric graphs with reservoirs, that is graphs where a one-dimensional interval is associated to each edge and, in addition, the vertices are able to store and exchange mass with these intervals. Focusing on…