Approximation of Functionals by Neural Network without Curse of Dimensionality

Abstract

In this paper, we establish a neural network to approximate functionals, which are maps from infinite dimensional spaces to finite dimensional spaces. The approximation error of the neural network is $O(1/\sqrt{m})$ where $m$ is the size of networks, which overcomes the curse of dimensionality. The key idea of the approximation is to define a Barron spectral space of functionals.

References in corpus (12)

• DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators
• DGM: A deep learning algorithm for solving partial differential equations
• Solving high-dimensional partial differential equations using deep learning
• Fourier Neural Operator for Parametric Partial Differential Equations
• Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations
• Breaking the Curse of Dimensionality with Convex Neural Networks
• Multipole Graph Neural Operator for Parametric Partial Differential Equations
• Two-Layer Neural Networks for Partial Differential Equations: Optimization and Generalization Theory
• Deep Learning with Functional Inputs
• MOD-Net: A Machine Learning Approach via Model-Operator-Data Network for Solving PDEs
• Deep Nonparametric Estimation of Operators between Infinite Dimensional Spaces
• Complexity Measures for Neural Networks with General Activation Functions Using Path-based Norms