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Exponential Expressivity of ReLU^k Neural Networks on Gevrey Classes with Point Singularities
(2024)SAM Research ReportWe analyze deep Neural Network emulation rates of smooth functions with point singularities in bounded, polytopal domains \(\mathrm{D} \subset \mathbb{R}^d\), \(d=2,3\). We prove exponential emulation rates in Sobolev spaces in terms of the number of neurons and in terms of the number of nonzero coefficients for Gevrey-regular solution classes defined in terms of weighted Sobolev scales in \(\mathrm{D}\), comprising the countably-normed ...Report -
Banach lattices with upper p-estimates: Free and injective objects
(2024)SAM Research ReportWe study the free Banach lattice FBL(p,∞)[E] with upper p-estimates generated by a Banach space E. Using a classical result of Pisier on factorization through Lp,∞(μ) together with a finite dimensional reduction, it is shown that the spaces ℓp,∞(n) witness the universal property of FBL(p,∞)[E] isomorphically. As a consequence, we obtain a functional representation for FBL(p,∞)[E], answering a previously open question. More generally, our ...Report -
Wavelet compressed, modified Hilbert transform in the space-time discretization of the heat equation
(2024)SAM Research ReportOn a finite time interval \((0,T)\), we consider the multiresolution Galerkin discretization of a modified Hilbert transform \((H_T)\) which arises in the space-time Galerkin discretization of the linear diffusion equation. To this end, we design spline-wavelet systems in \((0,T)\) consisting of piecewise polynomials of degree \(\geq 1\) with sufficiently many vanishing moments which constitute Riesz bases in the Sobolev spaces \( ...Report -
Time-dependent electromagnetic scattering from dispersive materials
(2024)SAM Research ReportThis paper studies time-dependent electromagnetic scattering from metamaterials that are described by dispersive material laws. We consider the numerical treatment of a scattering problem in which a dispersive material law, for a causal and passive homogeneous material, determines the wave-material interaction in the scatterer. The resulting problem is nonlocal in time inside the scatterer and is posed on an unbounded domain. Well-posedness ...Report -
The Language of Hyperelastic Materials
(2024)SAM Research ReportThe automated discovery of constitutive laws forms an emerging area that focuses on automatically obtaining symbolic expressions describing the constitutive behavior of solid materials from experimental data. Existing symbolic/sparse regression methods rely on availability of libraries of material models, which are typically hand-designed by a human expert relying on known models as reference, or deploy generative algorithms with exponential ...Report -
Efficient Computation of Large-Scale Statistical Solutions to Incompressible Fluid Flows
(2024)SAM Research ReportThis work presents the development, performance analysis and subsequent optimization of a GPU-based spectral hyperviscosity solver for turbulent flows described by the three dimensional incompressible Navier-Stokes equations. The method solves for the fluid velocity fields directly in Fourier space, eliminating the need to solve a large-scale linear system of equations in order to find the pressure field. Special focus is put on the ...Report -
Neural Networks for Singular Perturbations
(2024)SAM Research ReportWe prove deep neural network (DNN for short) expressivity rate bounds for solution sets of a model class of singularly perturbed, elliptic two-point boundary value problems, in Sobolev norms, on the bounded interval (−1,1). We assume that the given source term and reaction coefficient are analytic in [−1,1]. We establish expression rate bounds in Sobolev norms in terms of the NN size which are uniform with respect to the singular perturbation ...Report -
Numerical analysis of physics-informed neural networks and related models in physics-informed machine learning
(2024)SAM Research ReportPhysics-informed neural networks (PINNs) and their variants have been very popular in recent years as algorithms for the numerical simulation of both forward and inverse problems for partial differential equations. This article aims to provide a comprehensive review of currently available results on the numerical analysis of PINNs and related models that constitute the backbone of physics-informed machine learning. We provide a unified ...Report -
Exponentially localised interface eigenmodes in finite chains of resonators
(2024)SAM Research ReportThis paper studies wave localisation in chains of finitely many resonators. There is an extensive theory predicting the existence of localised modes induced by defects in infinitely periodic systems. This work extends these principles to finite-sized systems. We consider finite systems of subwavelength resonators arranged in dimers that have a geometric defect in the structure. This is a classical wave analogue of the Su-Schrieffer-Heeger ...Report -
Spectra and pseudo-spectra of tridiagonal k-Toeplitz matrices and the topological origin of the non-Hermitian skin effect
(2024)SAM Research ReportWe establish new results on the spectra and pseudo-spectra of tridiagonal k-Toeplitz operators and matrices. In particular, we prove the connection between the winding number of the eigenvalues of the symbol function and the exponential decay of the associated eigenvectors (or pseudo-eigenvectors). Our results elucidate the topological origin of the non-Hermitian skin effect in general one-dimensional polymer systems of subwavelength ...Report