Proof of the theory-to-practice gap in deep learning via sampling complexity bounds for neural network approximation spaces

DOI10.1007/S10208-023-09607-WMaRDI QIDQ6592113

Philipp Grohs, Felix Voigtlaender

Publication date: 24 August 2024

Published in: Foundations of Computational Mathematics (Search for Journal in Brave)

zbMATH Keywords

approximation spacesGelfand numbersrandomized approximationinformation based complexitydeep neural networkstheory-to-computational gaps

Mathematics Subject Classification ID

Artificial neural networks and deep learning (68T07) Abstract approximation theory (approximation in normed linear spaces and other abstract spaces) (41A65) Approximation by arbitrary nonlinear expressions; widths and entropy (41A46)

Cites Work

• The Deep Ritz Method: a deep learning-based numerical algorithm for solving variational problems
• Approximation spaces of deep neural networks
• Optimal approximation of piecewise smooth functions using deep ReLU neural networks
• A proof that rectified deep neural networks overcome the curse of dimensionality in the numerical approximation of semilinear heat equations
• Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations
• On the mathematical foundations of learning
• High-Dimensional Probability
• The Gap between Theory and Practice in Function Approximation with Deep Neural Networks
• Deep Neural Network Approximation Theory
• Optimal Approximation with Sparsely Connected Deep Neural Networks
• 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
• Solving inverse problems using data-driven models
• Understanding Machine Learning
• Neural network approximation
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Related Items (1)

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