A theoretical analysis of deep neural networks and parametric PDEs (Q2117329)

scientific article; zbMATH DE number 7493717

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English	A theoretical analysis of deep neural networks and parametric PDEs	scientific article; zbMATH DE number 7493717

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instance of

scholarly article

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title

A theoretical analysis of deep neural networks and parametric PDEs (English)

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Constructive Approximation

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publication date

21 March 2022

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full work available at URL

https://arxiv.org/abs/1904.00377

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zbMATH Keywords

deep neural networks

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parametric PDEs

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approximation rates

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reduced basis method

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describes a project that uses

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Kolmogorov widths and low-rank approximations of parametric elliptic PDEs

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Fully Discrete Approximation of Parametric and Stochastic Elliptic PDEs

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Parametric PDEs: sparse or low-rank approximations?

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Sparse polynomial approximation of parametric elliptic PDEs. Part I: affine coefficients

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Universal approximation bounds for superpositions of a sigmoidal function

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Analysis of the Generalization Error: Empirical Risk Minimization over Deep Artificial Neural Networks Overcomes the Curse of Dimensionality in the Numerical Approximation of Black--Scholes Partial Differential Equations

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Convergence Rates for Greedy Algorithms in Reduced Basis Methods

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Optimal Approximation with Sparsely Connected Deep Neural Networks

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A Posteriori Error Analysis of the Reduced Basis Method for Nonaffine Parametrized Nonlinear PDEs

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Breaking the curse of dimensionality in sparse polynomial approximation of parametric PDEs

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Approximation of high-dimensional parametric PDEs

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On the mathematical foundations of learning

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Approximation by superpositions of a sigmoidal function

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How To Best Sample a Solution Manifold?

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Data driven approximation of parametrized PDEs by reduced basis and neural networks

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Greedy algorithms for reduced bases in Banach spaces

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Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations

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Variational Monte Carlo -- bridging concepts of machine learning and high-dimensional partial differential equations

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Numerical solution of the parametric diffusion equation by deep neural networks

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Q5270493

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Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations

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Deep Neural Network Approximation Theory

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Error bounds for approximations with deep ReLU neural networks in Ws,p norms

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Solving high-dimensional partial differential equations using deep learning

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Relu Deep Neural Networks and Linear Finite Elements

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Certified Reduced Basis Methods for Parametrized Partial Differential Equations

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Non-intrusive reduced order modeling of nonlinear problems using neural networks

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Multilayer feedforward networks are universal approximators

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A proof that rectified deep neural networks overcome the curse of dimensionality in the numerical approximation of semilinear heat equations

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Proof that deep artificial neural networks overcome the curse of dimensionality in the numerical approximation of Kolmogorov partial differential equations with constant diffusion and nonlinear drift coefficients

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Reduced basis method for quadratically nonlinear transport equations

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Solving parametric PDE problems with artificial neural networks

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Model reduction of dynamical systems on nonlinear manifolds using deep convolutional autoencoders

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Lower bounds for approximation by MLP neural networks

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Approximation properties of a multilayered feedforward artificial neural network

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Recent advances in reduction methods for nonlinear problems

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On making large nonlinear problems small

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Deep ReLU networks and high-order finite element methods

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Optimal approximation of piecewise smooth functions using deep ReLU neural networks

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Equivalence of approximation by convolutional neural networks and fully-connected networks

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Reduced Basis Methods for Partial Differential Equations

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Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equations

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Rectified deep neural networks overcome the curse of dimensionality for nonsmooth value functions in zero-sum games of nonlinear stiff systems

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Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations. Application to transport and continuum mechanics.

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Deep learning in high dimension: Neural network expression rates for generalized polynomial chaos expansions in UQ

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Provable approximation properties for deep neural networks

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DGM: a deep learning algorithm for solving partial differential equations

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Gaussian elimination is not optimal

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Introduction to Uncertainty Quantification

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Error bounds for approximations with deep ReLU networks

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Multilevel approximation of parametric and stochastic PDES

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Identifiers

Mathematics Subject Classification ID

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10.1007/S00365-021-09551-4

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Sitelinks

Mathematics(1 entry)

mardi Publication:2117329