399 research outputs found
A weighted first-order formulation for solving anisotropic diffusion equations with deep neural networks
In this paper, a new weighted first-order formulation is proposed for solving
the anisotropic diffusion equations with deep neural networks. For many
numerical schemes, the accurate approximation of anisotropic heat flux is
crucial for the overall accuracy. In this work, the heat flux is firstly
decomposed into two components along the two eigenvectors of the diffusion
tensor, thus the anisotropic heat flux approximation is converted into the
approximation of two isotropic components.
Moreover, to handle the possible jump of the diffusion tensor across the
interface, the weighted first-order formulation is obtained by multiplying this
first-order formulation by a weighted function. By the decaying property of the
weighted function, the weighted first-order formulation is always well-defined
in the pointwise way. Finally, the weighted first-order formulation is solved
with deep neural network approximation. Compared to the neural network
approximation with the original second-order elliptic formulation, the proposed
method can significantly improve the accuracy, especially for the discontinuous
anisotropic diffusion problems
From Continuous Dynamics to Graph Neural Networks: Neural Diffusion and Beyond
Graph neural networks (GNNs) have demonstrated significant promise in
modelling relational data and have been widely applied in various fields of
interest. The key mechanism behind GNNs is the so-called message passing where
information is being iteratively aggregated to central nodes from their
neighbourhood. Such a scheme has been found to be intrinsically linked to a
physical process known as heat diffusion, where the propagation of GNNs
naturally corresponds to the evolution of heat density. Analogizing the process
of message passing to the heat dynamics allows to fundamentally understand the
power and pitfalls of GNNs and consequently informs better model design.
Recently, there emerges a plethora of works that proposes GNNs inspired from
the continuous dynamics formulation, in an attempt to mitigate the known
limitations of GNNs, such as oversmoothing and oversquashing. In this survey,
we provide the first systematic and comprehensive review of studies that
leverage the continuous perspective of GNNs. To this end, we introduce
foundational ingredients for adapting continuous dynamics to GNNs, along with a
general framework for the design of graph neural dynamics. We then review and
categorize existing works based on their driven mechanisms and underlying
dynamics. We also summarize how the limitations of classic GNNs can be
addressed under the continuous framework. We conclude by identifying multiple
open research directions
Innovative Approaches to the Numerical Approximation of PDEs
This workshop was about the numerical solution of PDEs for which classical
approaches,
such as the finite element method, are not well suited or need further
(theoretical) underpinnings.
A prominent example of PDEs for which classical methods are not well
suited are PDEs posed in high space dimensions.
New results on low rank tensor approximation for those problems were
presented.
Other presentations dealt with regularity of PDEs, the numerical solution
of PDEs on surfaces,
PDEs of fractional order, numerical solvers for PDEs that converge with
exponential rates, and
the application of deep neural networks for solving PDEs
A deep learning framework for multi-scale models based on physics-informed neural networks
Physics-informed neural networks (PINN) combine deep neural networks with the
solution of partial differential equations (PDEs), creating a new and promising
research area for numerically solving PDEs. Faced with a class of multi-scale
problems that include loss terms of different orders of magnitude in the loss
function, it is challenging for standard PINN methods to obtain an available
prediction. In this paper, we propose a new framework for solving multi-scale
problems by reconstructing the loss function. The framework is based on the
standard PINN method, and it modifies the loss function of the standard PINN
method by applying different numbers of power operations to the loss terms of
different magnitudes, so that the individual loss terms composing the loss
function have approximately the same order of magnitude among themselves. In
addition, we give a grouping regularization strategy, and this strategy can
deal well with the problem which varies significantly in different subdomains.
The proposed method enables loss terms with different magnitudes to be
optimized simultaneously, and it advances the application of PINN for
multi-scale problems
Generalized averaged Gaussian quadrature and applications
A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal
MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications
Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described
SOLID-SHELL FINITE ELEMENT MODELS FOR EXPLICIT SIMULATIONS OF CRACK PROPAGATION IN THIN STRUCTURES
Crack propagation in thin shell structures due to cutting is conveniently simulated
using explicit finite element approaches, in view of the high nonlinearity of the problem. Solidshell
elements are usually preferred for the discretization in the presence of complex material
behavior and degradation phenomena such as delamination, since they allow for a correct
representation of the thickness geometry. However, in solid-shell elements the small thickness
leads to a very high maximum eigenfrequency, which imply very small stable time-steps. A new
selective mass scaling technique is proposed to increase the time-step size without affecting
accuracy. New ”directional” cohesive interface elements are used in conjunction with selective
mass scaling to account for the interaction with a sharp blade in cutting processes of thin ductile
shells
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