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Graph-Coupled Oscillator Networks
(2022)SAM Research ReportWe propose Graph-Coupled Oscillator Networks (GraphCON), a novel framework for deep learning on graphs. It is based on discretizations of a second-order system of ordinary differential equations (ODEs), which model a network of nonlinear forced and damped oscillators, coupled via the adjacency structure of the underlying graph. The flexibility of our framework permits any basic GNN layer (e.g. convolutional or attentional) as the coupling ...Report -
Fünfter Bericht zum AKAD-Bildungsangebot zur Berufsmaturität II: Ergebnisse der Längsschnittbefragung von Studierenden
(2022)CES StudiesDieser Bericht präsentiert die Resultate der Längsschnittbefragung von Studierenden während ihrer Ausbildung am AKAD College. Dabei wurden alle Studierenden, die zwischen August 2018 und Februar 2020 ein AKAD-Ausbildungsangebot zur Berufsmaturität II starteten, zu mehreren Zeitpunkten ihrer Ausbildung befragt. Die dabei erhobenen Daten wurden durch administrative Daten von AKAD College ergänzt. Die Befragung der Studierenden erhob deren ...Report -
Boundary Integral Exterior Calculus
(2022)SAM Research ReportWe develop first-kind boundary integral equations for Hodge-Dirac and Hodge-Laplace operators associated with de Rham Hilbert complexes on compact Riemannian manifolds and Euclidean space. We show that the first-kind boundary integral operators associated with Hodge-Dirac and Hodge-Laplace boundary value problems posed on submanifolds with Lipschitz boundaries are Hodge-Dirac and Hodge-Laplace operators as well, but associated with a trace ...Report -
Subwavelength resonances in 1D high-contrast acoustic media
(2022)SAM Research ReportWe propose a mathematical theory of acoustic wave scattering in one-dimensional finite high-contrast media. The system considered is constituted of a finite alternance of high-contrast segments of arbitrary lengths and interdistances, called the ``resonators'', and a background medium. We prove the existence of subwavelength resonances, which are the counterparts of the well-known Minnaert resonances in 3D systems. Our main contribution ...Report -
Transmission properties of space-time modulated metamaterials
(2022)SAM Research ReportWe prove the possibility of achieving exponentially growing wave propagation in space-time modulated media and give an asymptotic analysis of the quasifrequencies in terms of the amplitude of the time modulation at the degenerate points of the folded band structure. Our analysis provides the first proof of existence of k-gaps in the band structures of space-time modulated systems of subwavelength resonators.Report -
Weak physics informed neural networks for approximating entropy solutions of hyperbolic conservation laws
(2022)SAM Research ReportPhysics informed neural networks (PINNs) require regularity of solutions of the underlying PDE to guarantee accurate approximation. Consequently, they may fail at approximating discontinuous solutions of PDEs such as nonlinear hyperbolic equations. To ameliorate this, we propose a novel variant of PINNs, termed as weak PINNs (wPINNs) for accurate approximation of entropy solutions of scalar conservation laws. wPINNs are based on approximating ...Report -
Sampling at twice the Nyquist rate in two frequency bins guarantees uniqueness in Gabor phase retrieval
(2022)SAM Research ReportWe demonstrate that a fourfold redundancy in the measurements is sufficient for uniqueness in sampled Gabor phase retrieval with bandlimited signals and thereby draw a parallel between the sampled Gabor phase retrieval problem and finite-dimensional phase retrieval problems. Precisely, we show that sampling at twice the Nyquist rate in two frequency bins guarantees uniqueness in Gabor phase retrieval for signals in the Paley-Wiener space.Report -
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Quasi-local and frequency robust preconditioners for the Helmholtz first-kind integral equations on the disk
(2022)SAM Research ReportWe propose preconditioners for the Helmholtz scattering problems by a planar, disk-shaped screen in R3. Those preconditioners are approximations of the square-roots of some partial differential operators acting on the screen. Their matrix-vector products involve only a few sparse system resolutions and can thus be evaluated cheaply in the context of iterative methods. For the Laplace equation (i.e. for the wavenumber k=0) with Dirichlet ...Report -
ReLU Neural Network Galerkin BEM
(2022)SAM Research ReportWe introduce Neural Network (NN for short) approximation architectures for the numerical solution of Boundary Integral Equations (BIEs for short). We exemplify the proposed NN approach for the boundary reduction of the potential problem in two spatial dimensions. We adopt a Galerkin formulation based approach, in polygonal domains with a finite number of straight sides. Trial spaces used in the Galerkin discretization of the BIEs are built ...Report