Approximation of Functionals by Neural Network without Curse of Dimensionality
This addresses a fundamental challenge in functional approximation for applications in fields like PDEs or machine learning, offering a novel theoretical breakthrough rather than an incremental improvement.
The paper tackles the problem of approximating functionals (maps from infinite to finite dimensional spaces) using neural networks, achieving an approximation error of O(1/√m) that overcomes the curse of dimensionality, where m is the network size.
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.