Deep Neural Network Approximation

Deep neural network (DNN) approximation focuses on understanding and improving the ability of DNNs to accurately represent complex functions, particularly in high dimensions where traditional methods struggle. Current research investigates optimal approximation rates for various function classes (e.g., oscillatory, Hölder-continuous, and those with specific symmetries) using different activation functions and network architectures, including multi-grade deep learning models and those incorporating gradient information. These advancements are crucial for improving the efficiency and accuracy of DNNs in diverse applications, ranging from solving partial differential equations and integral equations to image processing and scientific computing.

Papers

February 12, 2024

Score-based generative models break the curse of dimensionality in learning a family of sub-Gaussian probability distributions
Frank Cole, Yulong Lu
RSD Difference of Gaussian Score Based Generative Data Dimensionality Reversal Curse Single Parent Family Sub Gaussian Deep Neural Network Approximation Independent Rate

January 13, 2024

Deep Neural Network Solutions for Oscillatory Fredholm Integral Equations
Jie Jiang, Yuesheng Xu
Deep Neural Network Deep Learning Solution Deep Neural Network Approximation

November 8, 2023

Optimal Deep Neural Network Approximation for Korobov Functions with respect to Sobolev Norms
Yahong Yang, Yulong Lu
Deep Neural Network Average Approximation Simple Function Approximation Rate Approximation Property Deep Neural Network Approximation Sobolev Norm Function Approximator

July 13, 2023

Deep Network Approximation: Beyond ReLU to Diverse Activation Functions
Shijun Zhang, Jianfeng Lu, Hongkai Zhao
Deep Neural Network Activation Function ReLU Layer Deep Neural Network Approximation ReLU Regression

April 12, 2023

Deep neural network approximation of composite functions without the curse of dimensionality
Adrian Riekert
Deep Neural Network High Dimensional Data Dimensionality Reversal Curse Continuous Space Deep Neural Network Parameter Deep Neural Network Approximation Function Composition

March 29, 2023

Conductivity Imaging from Internal Measurements with Mixed Least-Squares Deep Neural Networks
Bangti Jin, Xiyao Li, Qimeng Quan, Zhi Zhou
Deep Neural Network High Dimensional Problem Internal Measurement Deep Neural Network Approximation Electrical Conductivity Conductivity Distribution Conductivity Imaging

November 30, 2022

Neural Network Representation of Time Integrators
Rainald Löhner, Harbir Antil
Neural Network Representation Numerical Integration Deep Neural Network Approximation Runge Kutta Mass Damper Numerical Integrator

November 8, 2022

Gradient-enhanced deep neural network approximations
Xiaodong Feng, Li Zeng
Deep Neural Network Uncertainty Quantification Gradient Based Efficient Uncertainty Estimation Deep Neural Network Approximation

August 18, 2022

Deep Neural Network Approximation of Invariant Functions through Dynamical Systems
Qianxiao Li, Ting Lin, Zuowei Shen
Dynamical System Universal Approximation Deep Neural Network Approximation Roto Translation Invariant Function Symmetric Function

July 15, 2022

Error analysis for deep neural network approximations of parametric hyperbolic conservation laws
Tim De Ryck, Siddhartha Mishra
Neural Network Generalization Error Numerical Experiment Conservation Law Error Analysis Approximation Error Deep Neural Network Approximation

July 12, 2022

Robust and efficient computation of retinal fractal dimension through deep approximation
Justin Engelmann, Ana Villaplana-Velasco, Amos Storkey, Miguel O. Bernabeu
Image Analysis Efficient Computation Fractal Dimension Deep Neural Network Approximation Retinal Feature

May 7, 2022

Deep learning approximations for non-local nonlinear PDEs with Neumann boundary conditions
Victor Boussange, Sebastian Becker, Arnulf Jentzen, Benno Kuckuck, Loïc Pellissier
Boundary Condition Deep Neural Network Approximation Neumann to Dirichlet Operator

January 11, 2022

Uniform Approximation with Quadratic Neural Networks
Ahmed Abdeljawad
Neural Network Average Approximation Deep Neural Network Approximation Approximation Capability H\"older Function

November 25, 2021

Error Bounds for a Matrix-Vector Product Approximation with Deep ReLU Neural Networks
Tilahun M. Getu
Deep Learning Machine Learning Error Bound Machine Learning Paradigm Rectified Linear Unit Deep Neural Network Approximation

November 15, 2021

Deep Network Approximation in Terms of Intrinsic Parameters
Zuowei Shen, Haizhao Yang, Shijun Zhang
Deep Learning Learnable Parameter Model Parameter Contentious Term Approximation Power Deep Neural Network Approximation

Deep Neural Network Approximation

Papers

Score-based generative models break the curse of dimensionality in learning a family of sub-Gaussian probability distributions

Deep Neural Network Solutions for Oscillatory Fredholm Integral Equations

Optimal Deep Neural Network Approximation for Korobov Functions with respect to Sobolev Norms

Deep Network Approximation: Beyond ReLU to Diverse Activation Functions

Deep neural network approximation of composite functions without the curse of dimensionality

Conductivity Imaging from Internal Measurements with Mixed Least-Squares Deep Neural Networks

Neural Network Representation of Time Integrators

Gradient-enhanced deep neural network approximations

Deep Neural Network Approximation of Invariant Functions through Dynamical Systems

Error analysis for deep neural network approximations of parametric hyperbolic conservation laws

Robust and efficient computation of retinal fractal dimension through deep approximation

Deep learning approximations for non-local nonlinear PDEs with Neumann boundary conditions

Uniform Approximation with Quadratic Neural Networks

Error Bounds for a Matrix-Vector Product Approximation with Deep ReLU Neural Networks

Deep Network Approximation in Terms of Intrinsic Parameters