623 research outputs found
A new nonlocal nonlinear diffusion equation for image denoising and data analysis
In this paper we introduce and study a new feature-preserving nonlinear
anisotropic diffusion for denoising signals. The proposed partial differential
equation is based on a novel diffusivity coefficient that uses a nonlocal
automatically detected parameter related to the local bounded variation and the
local oscillating pattern of the noisy input signal. We provide a mathematical
analysis of the existence of the solution of our nonlinear and nonlocal
diffusion equation in the two dimensional case (images processing). Finally, we
propose a numerical scheme with some numerical experiments which demonstrate
the effectiveness of the new method
Pseudo-Hamiltonian neural networks for learning partial differential equations
Pseudo-Hamiltonian neural networks (PHNN) were recently introduced for
learning dynamical systems that can be modelled by ordinary differential
equations. In this paper, we extend the method to partial differential
equations. The resulting model is comprised of up to three neural networks,
modelling terms representing conservation, dissipation and external forces, and
discrete convolution operators that can either be learned or be given as input.
We demonstrate numerically the superior performance of PHNN compared to a
baseline model that models the full dynamics by a single neural network.
Moreover, since the PHNN model consists of three parts with different physical
interpretations, these can be studied separately to gain insight into the
system, and the learned model is applicable also if external forces are removed
or changed.Comment: 33 pages, 14 figures; v2: minor changes to text, updated numerical
experiment
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