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Descriptive complexity of controllable graphs

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arXiv 2023年

作者： Abiad, Aida Dawar, Anuj Zapata, Octavio Department of Mathematics and Computer Science Eindhoven University of Technology Netherlands Department of Mathematics: Analysis Logic and Discrete Mathematics Ghent University Belgium Department of Mathematics and Data Science Vrije Universiteit Brussel Belgium Department of Computer Science and Technology University of Cambridge United Kingdom Instituto de Matemáticas Universidad Nacional Autónoma de México Mexico

Let G be a graph on n vertices with adjacency matrix A, and let 1 be the all-ones vector. We call G controllable if the set of vectors 1, A1, . . ., An−11 spans the whole space Rn. We characterize the isomorphism problem of controllable graphs in terms of other combinatorial, geometric and logical problems. We also describe a polynomial time algorithm for graph isomorphism that works for almost all graphs. © 2023, CC BY-SA.

关键词： Polynomial approximation

Constructing cospectral hypergraphs

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arXiv 2022年

作者： Abiad, Aida Khramova, Antonina P. Department of Mathematics and Computer Science Eindhoven University of Technology Netherlands Department of Mathematics: Analysis Logic and Discrete Mathematics Ghent University Belgium Department of Mathematics and Data Science Vrije Universiteit Brussel Belgium

Spectral hypergraph theory mainly concerns using hypergraph spectra to obtain structural information about the given hypergraphs. The study of cospectral hypergraphs is important since it reveals which hypergraph properties cannot be deduced from their spectra. In this paper, we show a new method for constructing cospectral uniform hypergraphs using two well-known hypergraph representations: adjacency tensors and adjacency matrices. © 2022, CC BY.

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Non-convergence to global minimizers in data driven supervised deep learning: Adam and stochastic gradient descent optimization provably fail to converge to global minimizers in the training of deep neural networks with ReLU activation

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arXiv 2024年

作者： Hannibal, Sonja Jentzen, Arnulf Thang, Do Minh Applied Mathematics: Institute for Analysis and Numerics Faculty of Mathematics and Computer Science University of Münster Germany School of Data Science Shenzhen Research Institute of Big Data The Chinese University of Hong Kong Shenzhen [CUHK-Shenzhen China School of Data Science The Chinese University of Hong Kong Shenzhen [CUHK-Shenzhen China Department of Probability and Statistic Institute of Mathematics Vietnam Academy of Science and Technology Viet Nam

Deep learning (DL) methods – consisting of a class of deep neural networks (DNNs) trained by a stochastic gradient descent (SGD) optimization method – are nowadays key tools to solve data driven supervised learning problems. Despite the great success of SGD methods in the training of DNNs, it remains a fundamental open problem of research to explain the success and the limitations of such methods in rigorous theoretical terms. In particular, even in the standard setup of data driven supervised learning problems, it remained an open research problem to prove (or disprove) that SGD methods converge in the training of DNNs with the popular rectified linear unit (ReLU) activation function with high probability to global minimizers in the optimization landscape. In this work we answer this question negatively by proving that it does not hold that SGD methods converge with high probability to global minimizers of the objective function. Even stronger, in this work we prove for a large class of SGD methods that the considered optimizer does with high probability not converge to global minimizers of the optimization problem. It turns out that the probability to not converge to a global minimizer converges at least exponentially quickly to one as the width of the first hidden layer of the ANN (the number of neurons on the first hidden layer) and the depth of the ANN (the number of hidden layers), respectively, increase to infinity. The general non-convergence results of this work do not only apply to the plain vanilla standard SGD method but also to a large class of accelerated and adaptive SGD methods such as the momentum SGD, the Nesterov accelerated SGD, the Adagrad, the RMSProp, the Adam, the Adamax, the AMSGrad, and the Nadam optimizers. However, we would like to emphasize that the findings of this work do not imply that SGD methods do not succeed to train DNNs: it may still very well be the case that SGD methods provably succeed to train DNNs in data driven learning pr

关键词： Self-supervised learning

The necessity of depth for artificial neural networks to approximate certain classes of smooth and bounded functions without the curse of dimensionality

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arXiv 2023年

作者： Gonon, Lukas Graeber, Robin Jentzen, Arnulf Faculty of Natural Sciences Department of Mathematics Imperial College London United Kingdom Institute for Analysis and Numerics Faculty of Mathematics and Computer Science University of Münster Germany School of Data Science and Shenzhen Research Institute of Big Data The Chinese University of Hong Kong Shenzhen China

In this article we study high-dimensional approximation capacities of shallow and deep artificial neural networks (ANNs) with the rectified linear unit (ReLU) activation. In particular, it is a key contribution of this work to reveal that for all a, b ∈ with b−a ≥ 7 we have that the functions (Formula Presented) as well as the functions (formula Presented) can neither be approximated without the curse of dimensionality by means of shallow ANNs nor insufficiently deep ANNs with ReLU activation but can be approximated without the curse of dimensionality by sufficiently deep ANNs with ReLU activation. More specifically, we prove that in the case of shallow ReLU ANNs or deep ReLU ANNs with a fixed number of hidden layers and with the size of scalar real parameters of the approximating ANNs growing at most polynomially in the dimension d ∈ N we have that the number of ANN parameters must grow at least exponentially in the dimension d ∈ N while in the case of deep ReLU ANNs with the number of hidden layers growing in the dimension d ∈ N we have that the number of scalar real parameters of the approximating ANNs grows at most polynomially in both the inverse of the prescribed approximation accuracy Ε > 0 and the dimension d ∈ , even if the absolute values of the ANN parameters are assumed to be uniformly bounded by one. We thus show that the product functions and the sine of the product functions are polynomially tractable approximation problems among the approximating class of deep ReLU ANNs with the number of hidden layers being allowed to grow in the dimension d ∈ . We establish the above outlined statement not only for the product functions and the sine of the product functions but also for other classes of target functions, in particular, for classes of uniformly globally bounded C∞-functions with compact support on any [a, b]d with a ∈ , b ∈ (a, ∞). Roughly speaking, in this work we lay open that simple approximation problems such as approximating the sine or cosine

关键词： Chemical activation

The M-matrix group inverse problem for distance-biregular graphs

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arXiv 2022年

作者： Abiad, Aida Carmona, Ángeles Encinas, Andrés M. Jiménez, María José Department of Mathematics and Computer Science Eindhoven University of Technology Netherlands Department of Mathematics: Analysis Logic and Discrete Mathematics Ghent University Belgium Department of Mathematics and Data Science Vrije Universiteit Brussel Belgium Department of Mathematics Polytechnic University of Catalonia Spain

In this paper we provide the group inverse of the combinatorial Laplacian matrix of distance–biregular graphs using the so–called equilibrium measures for sets obtained by deleting a vertex. We also show that the two equilibrium arrays characterizing a distance–biregular graph can be expressed in terms of the mentioned equilibrium measures. As a consequence of the minimum principle, we show a characterization of when the group inverse of the combinatorial Laplacian matrix of a distance–biregular graph is an M–matrix. © 2022, CC BY.

关键词： Inverse problems

Set-Valued Markov Chain Dependability Model with Uncertain data

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Set-Valued Markov Chain Dependability Model with Uncertain D...

IEEE International Conference on Dependable Systems, Services and Technologies (DESSERT)

作者： Leonid Lyubchyk Yurii Zaitsev Galyna Grinberg Olga Kostyuk Department of Computer Mathematics and Data Analysis National Technical University “KhPI” Kharkiv Ukraine Department of Markening National Technical University “KhPI” Kharkiv Ukraine

ISBN: (纸本)9798350333053

The development of Markov model approach for probabilistic analysis of dependable systems with uncertain data is considered. Further development of Markov models with interval description of uncertain parameters based on optimization approach to obtaining estimates and ellipsoidal approximations of transition matrices is proposed, which makes it possible to simplify the computational procedures for obtaining boundary estimates for transition and steady-state probabilities of the systems under study. Using the method of Lagrange functions, solutions of optimization problems for obtaining interval estimates of state probabilities are proposed. The proposed technique is illustrated by analysis of cyber-physical system under parametric uncertainty.

关键词： Analytical models Uncertainty Computational modeling Markov processes Probabilistic logic data models Steady-state

An Overview on Machine Learning Methods for Partial Differential Equations: From Physics Informed Neural Networks to Deep Operator Learning

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arXiv 2024年

作者： Gonon, Lukas Jentzen, Arnulf Kuckuck, Benno Liang, Siyu Riekert, Adrian von Wurstemberger, Philippe Department of Mathematics Imperial College London United Kingdom School of Data Science Shenzhen Research Institute of Big Data The Chinese University of Hong Kong Shenzhen [CUHK-Shenzhen China Applied Mathematics: Institute for Analysis and Numerics Faculty of Mathematics and Computer Science University of Münster Germany School of Mathematics and Statistics Nanjing University of Science and Technology Nanjing China School of Data Science The Chinese University of Hong Kong Shenzhen [CUHK-Shenzhen China Mathematisches Institut Ludwig-Maximilians-Universität München Germany Risklab Department of Mathematics ETH Zurich Switzerland

The approximation of solutions of partial differential equations (PDEs) with numerical algorithms is a central topic in applied mathematics. For many decades, various types of methods for this purpose have been developed and extensively studied. One class of methods which has received a lot of attention in recent years are machine learning-based methods, which typically involve the training of artificial neural networks (ANNs) by means of stochastic gradient descent type optimization methods. While approximation methods for PDEs using ANNs have first been proposed in the 1990s they have only gained wide popularity in the last decade with the rise of deep learning. This article aims to provide an introduction to some of these methods and the mathematical theory on which they are based. We discuss methods such as physics-informed neural networks (PINNs) and deep BSDE methods and consider several operator learning approaches. Copyright © 2024, The Authors. All rights reserved.

关键词： Partial differential equations

Machine Learning-Based Failure Rate Identification for Predictive Maintenance in Industry 4.0

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Machine Learning-Based Failure Rate Identification for Predi...

IEEE International Conference on Dependable Systems, Services and Technologies (DESSERT)

作者： Leonid Lyubchyk Olena Akhiiezer Galyna Grinberg Klym Yamkovyi Department of Computer Mathematics and Data Analysis National Technical University “KhPI” Kharkiv Ukraine Department of Markening National Technical University “KhPI” Kharkiv Ukraine

ISBN: (纸本)9798350333053

The problem of regression model of failure rate building using datasets containing a number of failures of recoverable systems at certain time intervals and measurements of influencing technological and operational factors is considered. To solve the problem, kernel method of machine learning along with ridge regression is used, which providing possibility of efficient approximation of failure rate dependence from a large number of influencing technological and operational factors, without the need to pre-select model structure, complexity of which is determined by amount of available training data. The technique is generalized to the problem of non-stationary failure rate prediction, the model of which is trained on a sliding data window. The proposed approach is illustrated by solving the problem of failure rate model building for semiconductor production equipment using real data.

关键词： Semiconductor device modeling Analytical models Buildings Training data Predictive models Big data data models

Improving hp-Variational Physics-Informed Neural Networks for Steady-State Convection-Dominated Problems

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arXiv 2024年

作者： Anandh, Thivin Ghose, Divij Jain, Himanshu Sunkad, Pratham Ganesan, Sashikumaar John, Volker Department of Computational and Data Sciences Indian Institute of Science Karnataka Bangalore India Department of Mechanical Engineering Indian Institute of Technology Tamil Nadu Madras India Weierstrass Institute for Applied Analysis and Stochastics Berlin Germany Department of Mathematics and Computer Science Freie University Berlin Germany

This paper proposes and studies two extensions of applying hp-variational physics-informed neural networks, more precisely the FastVPINNs framework, to convection-dominated convection-diffusion-reaction problems. First, a term in the spirit of a SUPG stabilization is included in the loss functional and a network architecture is proposed that predicts spatially varying stabilization parameters. Having observed that the selection of the indicator function in hard-constrained Dirichlet boundary conditions has a big impact on the accuracy of the computed solutions, the second novelty is the proposal of a network architecture that learns good parameters for a class of indicator functions. Numerical studies show that both proposals lead to noticeably more accurate results than approaches that can be found in the literature. © 2024, CC BY-NC-SA.

关键词： Stabilization