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

DOI10.1142/S0219530519410021zbMath1452.41009arXiv1902.07896OpenAlexW2969750612WikidataQ127354303 ScholiaQ127354303MaRDI QIDQ5132228

Gitta Kutyniok, Ingo Gühring, Philipp Petersen

Publication date: 10 November 2020

Published in: Analysis and Applications (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1902.07896

zbMATH Keywords

Sobolev spacesPDEsapproximation ratescurse of dimensiondeep neural networks

Mathematics Subject Classification ID

Learning and adaptive systems in artificial intelligence (68T05) Sobolev spaces and other spaces of ``smooth functions, embedding theorems, trace theorems (46E35) Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs (65N30) Rate of convergence, degree of approximation (41A25)

Related Items (45)

On sharpness of an error bound for deep ReLU network approximation ⋮ A deep learning approach to Reduced Order Modelling of parameter dependent partial differential equations ⋮ Deep learning for the partially linear Cox model ⋮ Approximation properties of deep ReLU CNNs ⋮ ReLU deep neural networks from the hierarchical basis perspective ⋮ Variational physics informed neural networks: the role of quadratures and test functions ⋮ Approximations with deep neural networks in Sobolev time-space ⋮ Convergence Rate Analysis for Deep Ritz Method ⋮ Full error analysis for the training of deep neural networks ⋮ Simultaneous neural network approximation for smooth functions ⋮ On the approximation of functions by tanh neural networks ⋮ Deep ReLU neural network approximation in Bochner spaces and applications to parametric PDEs ⋮ Rates of approximation by ReLU shallow neural networks ⋮ Deep learning methods for partial differential equations and related parameter identification problems ⋮ Some elliptic second order problems and neural network solutions: existence and error estimates ⋮ Neural control of discrete weak formulations: Galerkin, least squares \& minimal-residual methods with quasi-optimal weights ⋮ On decision regions of narrow deep neural networks ⋮ Deep ReLU neural networks in high-dimensional approximation ⋮ Exponential ReLU neural network approximation rates for point and edge singularities ⋮ Friedrichs Learning: Weak Solutions of Partial Differential Equations via Deep Learning ⋮ Neural ODE Control for Classification, Approximation, and Transport ⋮ SignReLU neural network and its approximation ability ⋮ Deep Neural Networks with ReLU-Sine-Exponential Activations Break Curse of Dimensionality in Approximation on Hölder Class ⋮ DPK: Deep Neural Network Approximation of the First Piola-Kirchhoff Stress ⋮ A Novel Deep Neural Network Algorithm for the Helmholtz Scattering Problem In the Unbounded Domain ⋮ Collocation approximation by deep neural ReLU networks for parametric and stochastic PDEs with lognormal inputs ⋮ DeepBND: a machine learning approach to enhance multiscale solid mechanics ⋮ Deep learning via dynamical systems: an approximation perspective ⋮ Growing axons: greedy learning of neural networks with application to function approximation ⋮ Neural Control of Parametric Solutions for High-Dimensional Evolution PDEs ⋮ Learning elliptic partial differential equations with randomized linear algebra ⋮ Simultaneous approximation of a smooth function and its derivatives by deep neural networks with piecewise-polynomial activations ⋮ Deep Network Approximation for Smooth Functions ⋮ Deep Nitsche Method: Deep Ritz Method with Essential Boundary Conditions ⋮ Non-Intrusive Reduced Order Modeling of Convection Dominated Flows Using Artificial Neural Networks with Application to Rayleigh-Taylor Instability ⋮ Unnamed Item ⋮ Efficient approximation of solutions of parametric linear transport equations by ReLU DNNs ⋮ Unnamed Item ⋮ Approximation rates for neural networks with encodable weights in smoothness spaces ⋮ The Gap between Theory and Practice in Function Approximation with Deep Neural Networks ⋮ Solving PDEs by variational physics-informed neural networks: an a posteriori error analysis ⋮ Deep Network With Approximation Error Being Reciprocal of Width to Power of Square Root of Depth ⋮ Error analysis for physics-informed neural networks (PINNs) approximating Kolmogorov PDEs ⋮ Optimization with learning-informed differential equation constraints and its applications ⋮ A theoretical analysis of deep neural networks and parametric PDEs

Uses Software

• ImageNet
• AlexNet
• DGM

Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Nonlocal diffusion and applications
• Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations
• An introduction to Sobolev spaces and interpolation spaces
• Functional analysis, Sobolev spaces and partial differential equations
• Approximation and estimation bounds for artificial neural networks
• Elliptic partial differential equations of second order
• Provable approximation properties for deep neural networks
• The Deep Ritz Method: a deep learning-based numerical algorithm for solving variational problems
• On the degree of nonlinear spline approximation in Besov-Sobolev spaces
• DGM: a deep learning algorithm for solving partial differential equations
• Optimal approximation of piecewise smooth functions using deep ReLU neural networks
• Error bounds for approximations with deep ReLU networks
• Group Invariant Scattering
• Learning Theory
• Deep learning in high dimension: Neural network expression rates for generalized polynomial chaos expansions in UQ
• Interpolation Theory
• Neural Network Learning
• Solving high-dimensional partial differential equations using deep learning
• Optimal Approximation with Sparsely Connected Deep Neural Networks
• Deep neural networks for rotation-invariance approximation and learning
• The Mathematical Theory of Finite Element Methods
• Nonlinear partial differential equations with applications
• Approximation by superpositions of a sigmoidal function

This page was built for publication: Error bounds for approximations with deep ReLU neural networks in Ws,p norms