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Stable recovery of entangled weights: Towards robust identification of deep neural networks from minimal samples

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arXiv 2021年

作者： Fiedler, Christian Fornasier, Massimo Klock, Timo Rauchensteiner, Michael Intelligent Control Systems Group Max Planck Institute for Intelligent Systems Heisenbergstr. 3 Stuttgart70569 Germany Institute for Data Science in Mechanical Engineering RWTH-Aachen University Dennewartstraße 27 Aachen50868 Germany Department of Mathematics Bolzmannstraße 3 85748 Garching Germany Department of Numerical Analysis and Scientific Computing Simula Research Laboratory Oslo Norway Department of Mathematics University of San Diego California San Diego United States

In this paper we approach the problem of unique and stable identifiability of generic deep artificial neural networks with pyramidal shape and smooth activation functions from a finite number of input-output samples. More specifically we introduce the so-called entangled weights, which compose weights of successive layers intertwined with suitable diagonal and invertible matrices depending on the activation functions and their shifts. We prove that entangled weights are completely and stably approximated by an efficient and robust algorithm as soon as O(D2 × m) nonadaptive input-output samples of the network are collected, where D is the input dimension and m is the number of neurons of the network. Moreover, we empirically observe that the approach applies to networks with up to O(D × mL) neurons, where mL is the number of output neurons at layer L. Provided knowledge of layer assignments of entangled weights and of remaining scaling and shift parameters, which may be further heuristically obtained by least squares, the entangled weights identify the network completely and uniquely. To highlight the relevance of the theoretical result of stable recovery of entangled weights, we present numerical experiments, which demonstrate that multilayered networks with generic weights can be robustly identified and therefore uniformly approximated by the presented algorithmic pipeline. In contrast backpropagation cannot generalize stably very well in this setting, being always limited by relatively large uniform error. In terms of practical impact, our study shows that we can relate input-output information uniquely and stably to network parameters, providing a form of explainability. Moreover, our method paves the way for compression of overparametrized networks and for the training of minimal complexity *** Codes 65D15, 68T07, 90C26 Copyright © 2021, The Authors. All rights reserved.

关键词： Deep neural networks

A polynomial-degree-robust a posteriori error estimator for nédélec discretizations of magnetostatic problems

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arXiv 2020年

作者： Gedicke, Joscha Geevers, Sjoerd Perugia, Ilaria Schöberl, Joachim Institute for Numerical Simulation University of Bonn Endenicher Allee 19b Bonn53115 Germany Faculty of Mathematics University of Vienna Oskar-Morgenstern-Platz 1 Vienna1090 Austria Institute for Analysis and Scientific Computing Vienna University of Technology Wiedner Hauptstrasse 8-10 Vienna1040 Austria

We present an equilibration-based a posteriori error estimator for Nédélec element discretizations of the magnetostatic problem. The estimator is obtained by adding a gradient correction to the estimator for Nédélec elements of arbitrary degree presented in [16]. This new estimator is proven to be reliable, with reliability constant 1, and efficient, with an efficiency constant that is independent of the polynomial degree of the approximation. These properties are demonstrated in a series of numerical experiments on three-dimensional test *** Codes 65N15, 65N30, 65N50 Copyright © 2020, The Authors. All rights reserved.

关键词： Magnetostatics

Twofold saddle-point formulation of biot poroelasticity with stress-dependent diffusion

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arXiv 2021年

作者： Gómez-Vargas, Bryan Mardal, Kent-André Ruiz-Baier, Ricardo Vinje, Vegard Sección de Matemática Sede de Occidente Universidad de Costa Rica San Ramón Ala juela Costa Rica Department of Mathematics Division of Mechanics University of Oslo Norway Simula Research Laboratory Lysaker1325 Norway School of Mathematics Monash University 9 Rainforest Walk MelbourneVIC3800 Australia Institute of Computer Science and Mathematical Modelling Sechenov University Moscow Russia Universidad Adventista de Chile Casilla 7-D Chillán Chile Department for Numerical Analysis and Scientific Computing Simula Research Laboratory Lysaker Norway

We introduce a stress/total-pressure formulation for poroelasticity that includes the coupling with steady nonlinear diffusion modified by stress. The nonlinear problem is written in mixed-primal form, coupling a perturbed twofold saddle-point system with an elliptic problem. The continuous formulation is analysed in the framework of abstract fixed-point theory and Fredholm alternative for compact operators. A mixed finite element method is proposed and its stability and convergence analysis are carried out. We also include a few illustrative numerical tests. The resulting model can be used to study waste removal in the brain parenchyma, where diffusion of a tracer alone or a combination of advection and diffusion are not sufficient to explain the alterations in rates of filtration observed in porous media *** Codes 65N60, 92C50 © 2021, CC BY.

关键词： Diffusion

Feasibility of Acousto-Electric Tomography

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arXiv 2019年

作者： Jensen, Bjørn Kirkeby, Adrian Knudsen, Kim Faculty of Information Technology University of Jyväskylä Finland Department of Numerical Analysis and Scientific Computing Simula Research Laboratory Oslo Norway Department of Applied Mathematics and Computer Science Technical University of Denmark Kgs Lyngby Denmark

In acousto-electric tomography the goal is to reconstruct the electric conductivity in a domain from electrostatic boundary measurements of corresponding currents and voltages, while the domain is penetrated by a time-dependent acoustic wave. In this paper we provide an in-silico model of the entire coupled physics scenario and perform computational experiments. We propose a complete inversion framework for acousto-electric tomography in two steps: First the interior power density is obtained from boundary measurements by solving a linear inverse problem;second the interior conductivity is reconstructed from the power density by solving a non-linear, fairly well-posed problem. We develop a numerical model with realistically chosen parameters inspired by medical imaging. The critical signal strength is analyzed and the omnipresent Johnson-Nyquist noise is estimated. With this setup we perform numerical experiments on synthetic data with noise and demonstrate the feasibility of the method. However, our findings are a mix of positive and negative. We reconstruct features even under severe noise conditions, but we also find that the required Signal-to-Noise ratio yet remains infeasible for practical *** Codes 35R30, 65N21 Copyright © 2019, The Authors. All rights reserved.

关键词： Inverse problems

Parametric sequential method for MRI-based wall shear stress quantification

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arXiv 2020年

作者： Shokina, Nina Teschner, Gabriel Bauer, Andreas Tropea, Cameron Egger, Herbert Hennig, Jürgen Krafft, Axel J. Department of Radiology Medical Physics Medical Center University of Freiburg Faculty of Medicine University of Freiburg Freiburg79106 Germany Institute for Numerical Analysis and Scientific Computing Department of Mathematics Technische Universität Darmstadt Darmstadt64293 Germany Institute for Fluid Mechanics and Aerodynamics Department of Mechanical Engineering Technische Universität Darmstadt Darmstadt-Griesheim64347 Germany Department of Radiology Medical Physics Medical Center University of Freiburg Faculty of Medicine University of Freiburg Freiburg79106 Germany

Wall shear stress (WSS) has been suggested as a potential biomarker in various cardiovascular diseases and it can be estimated from phase-contrast Magnetic Resonance Imaging (PC-MRI) velocity measurements. We present a new parametric sequential method for MRI-based WSS quantification consisting of a geometry identification and a subsequent approximation of the velocity field. The new method was validated on well controlled, high resolution in vitro measurements of turbulent stationary flows and physiological pulsatile flows in phantoms. Basic applicability to in vivo was demonstrated using 2D PC-MRI data of the ascending aorta of three volunteers. Copyright © 2020, The Authors. All rights reserved.

关键词： Magnetic resonance imaging

Automatic Variationally Stable analysis for Finite Element Computations: Transient Convection-Diffusion Problems

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arXiv 2020年

作者： Valseth, Eirik Behnoudfar, Pouria Dawson, Clint Romkes, Albert Oden Institute for Computational Engineering and Sciences University of Texas at Austin AustinTX78712 United States The Department of Data Science The Norwegian University of Life Science Drøbakveien 31 Ås1433 Norway Department of Scientific Computing and Numerical Analysis Simula Research Laboratory Kristian Augusts gate 23 Oslo0164 Norway PerthWA6152 Australia Department of Mechanical Engineering South Dakota School of Mines & Technology Rapid CitySD57701 United States

We present an application of stable finite element (FE) approximations of convection-diffusion initial boundary value problems (IBVPs) using a weighted least squares FE method, the automatic variationally stable finite element (AVS-FE) method [1]. The transient convection-diffusion problem leads to issues in classical FE methods as the differential operator can be considered a singular perturbation in both space and time. The stability property of the AVS-FE method, allows us significant flexibility in the construction of FE approximations in both space and time. Thus, in this paper, we take two distinct approaches to the FE discretization of the convection-diffusion problem: i) considering a space-time approach in which the temporal discretization is established using finite elements, and ii) a method of lines approach in which we employ the AVS-FE method in space whereas the temporal domain is discretized using the generalized-α method. We also consider another space-time technique in which the temporal direction is partitioned, thereby leading to finite space-time"slices" in an attempt to reduce the computational cost of the space-time discretizations. We present numerical verifications for these approaches, including numerical asymptotic convergence studies highlighting optimal convergence properties. Furthermore, in the spirit of the discontinuous Petrov-Galerkin (DPG) method by Demkowicz and Gopalakrishnan [2–6], the AVS-FE method also leads to readily available a posteriori error estimates through a Riesz representer of the residual of the AVS-FE approximations. Hence, the norm of the resulting local restrictions of these estimates serve as error indicators in both space and time for which we present multiple numerical verifications in mesh adaptive *** Codes 65M60 65M12 65M20 65M50 Copyright © 2020, The Authors. All rights reserved.

关键词： Finite element method

Finite element simulation of ionic electrodiffusion in cellular geometries

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arXiv 2019年

作者： Ellingsrud, A.J. Solbrå, A. Einevoll, G.T. Halnes, G. Rognes, M.E. Department for Scientific Computing and Numerical Analysis Simula Research Laboratory Norway Centre for Integrative Neuroplasticity University of Oslo Oslo Norway Department of Physics University of Oslo Oslo Norway Faculty of Science and Technology Norwegian University of Life Sciences Ås Norway

Mathematical models for excitable cells are commonly based on cable theory, which considers a homogenized domain and spatially constant ionic concentrations. Although such models provide valuable insight, the effect of altered ion concentrations or detailed cell morphology on the electrical potentials cannot be captured. In this paper, we discuss an alternative approach to detailed modelling of electrodiffusion in neural tissue. The mathematical model describes the distribution and evolution of ion concentrations in a geometrically-explicit representation of the intra- and extracellular domains. As a combination of the electroneutral Kirchhoff-Nernst-Planck (KNP) model and the Extracellular-Membrane-Intracellular (EMI) framework, we refer to this model as the KNP-EMI model. Here, we introduce and numerically evaluate a new, finite element-based numerical scheme for the KNP-EMI model, capable of efficiently and flexibly handling geometries of arbitrary dimension and arbitrary polynomial degree. Moreover, we compare the electrical potentials predicted by the KNP-EMI and EMI models. Finally, we study ephaptic coupling induced in an unmyelinated axon bundle and demonstrate how the KNP-EMI framework can give new insights in this setting. Copyright © 2019, The Authors. All rights reserved.

关键词： Finite element method

Hybrid discontinuous Galerkin methods with relaxed H(DIV)-conformity for incompressible flows. Part II

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arXiv 2018年

The present work is the second part of a pair of papers, considering Hybrid Discontinuous Galerkin methods with relaxed H(div)-conformity. The first part mainly dealt with presenting a robust analysis with respect to the mesh size h and the introduction of a reconstruction operator to restore divergence-conformity and pressure robustness using a non conforming right hand side. The aim of this part is the presentation of a high order polynomial robust analysis for the relaxed H(div)-conforming Hybrid Discontinuous Galerkin discretization of the two dimensional Stokes problem. It is based on the recently proven polynomial robust LBB-condition for BDM elements [P. L. Lederer, J. Schöberl, IMA Journal of numerical analysis, 2017] and is derived by a direct approach instead of using a best approximation Céa like result. We further treat the impact of the reconstruction operator on the hp analysis and present a numerical investigation considering polynomial robustness. We conclude the paper presenting an efficient operator splitting time integration scheme for the Navier–Stokes equations which is based on the methods recently presented in [C. Lehrenfeld, J. Schöberl, Comp. Meth. Appl. Mech. Eng., 361 (2016)] and includes the ideas of the reconstruction operator.35Q30, 65N12, 65N22, 65N30 Copyright © 2018, The Authors. All rights reserved.

关键词： Navier Stokes equations

Efficient white noise sampling and coupling for multilevel Monte Carlo with non-nested meshes

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arXiv 2018年

作者： Croci, Matteo Giles, Michael B. Rognes, Marie E. Farrell, Patrick E. Mathematical Institute University of Oxford Oxford United Kingdom Department for Numerical Analysis and Scientific Computing Simula Research Laboratory P.O. Box 134 Lysaker1325 Norway

When solving stochastic partial differential equations (SPDEs) driven by additive spatial white noise, the efficient sampling of white noise realizations can be challenging. Here, we present a new sampling technique that can be used to efficiently compute white noise samples in a finite element method and multilevel Monte Carlo (MLMC) setting. The key idea is to exploit the finite element matrix assembly procedure and factorize each local mass matrix independently, hence avoiding the factorization of a large matrix. Moreover, in a MLMC framework, the white noise samples must be coupled between subsequent levels. We show how our technique can be used to enforce this coupling even in the case of non-nested mesh hierarchies. We demonstrate the efficacy of our method with numerical experiments. We observe optimal convergence rates for the finite element solution of the elliptic SPDEs of interest in 2D and 3D and we show convergence of the sampled field covariances. In a MLMC setting, a good coupling is enforced and the telescoping sum is *** Codes 65C05, 60G60, 65N30, 60H35, 35R60 Copyright © 2018, The Authors. All rights reserved.

关键词： Partial differential equations