Sobolev Space

Sobolev spaces are mathematical frameworks used to analyze functions based on their smoothness and differentiability, finding applications in diverse fields like scientific computing and machine learning. Current research focuses on efficiently approximating functions within these spaces using deep neural networks, particularly exploring the effectiveness of architectures like ReLU networks and the impact of incorporating sensitivity information (Sobolev training) to improve model accuracy and convergence speed in applications such as solving partial differential equations. This work is significant because it improves the accuracy and efficiency of numerical methods for solving complex problems in areas like computational mechanics and PDE-based modeling, leading to advancements in fields ranging from engineering design to scientific simulations.

Papers

September 2, 2024

On the optimal approximation of Sobolev and Besov functions using deep ReLU neural networks
Yunfei Yang
Neural Network Average Approximation Function Space Approximation Rate Sparse Vector Sobolev Space

July 23, 2024

Sobolev neural network with residual weighting as a surrogate in linear and non-linear mechanics
A. O. M. Kilicsoy, J. Liedmann, M. A. Valdebenito, F. -J. Barthold, M. G. R. Faes
Neural Network Surrogate Model Dynamic Reweighting Solid Mechanic Surrogate Assisted Version Sobolev Training Sobolev Space

January 12, 2023

Learning Partial Differential Equations by Spectral Approximates of General Sobolev Spaces
Juan-Esteban Suarez Cardona, Phil-Alexander Hofmann, Michael Hecht
Partial Differential Equation Dimensional Approximation Inverse PDE Problem Spectral Approximation Sobolev Space

November 25, 2022

Optimal Approximation Rates for Deep ReLU Neural Networks on Sobolev and Besov Spaces
Jonathan W. Siegel
ReLU Network Function Space Deep ReLU Network \Ell_p$ Norm Approximation Rate Sharp Bound Sobolev Space

November 7, 2022

Inpainting in discrete Sobolev spaces: structural information for uncertainty reduction
Marco Seracini, Stephen R. Brown
Structural Information Exemplar Based Statistical Functionals Sobolev Space

May 15, 2022

Sobolev Acceleration and Statistical Optimality for Learning Elliptic Equations via Gradient Descent
Yiping Lu, Jose Blanchet, Lexing Ying
Gradient Descent Partial Differential Equation Reproducing Kernel Hilbert Space Sobolev Training Sobolev Norm Sobolev Space

Sobolev Space

Papers

On the optimal approximation of Sobolev and Besov functions using deep ReLU neural networks

Sobolev neural network with residual weighting as a surrogate in linear and non-linear mechanics

Learning Partial Differential Equations by Spectral Approximates of General Sobolev Spaces

Optimal Approximation Rates for Deep ReLU Neural Networks on Sobolev and Besov Spaces

Inpainting in discrete Sobolev spaces: structural information for uncertainty reduction

Sobolev Acceleration and Statistical Optimality for Learning Elliptic Equations via Gradient Descent