Session I.1 - Multiresolution and Adaptivity in Numerical PDEs
Poster
Approximation of Functionals by Neural Network without Curse of Dimensionality
Tianyu Jin
The Hong Kong University of Science and Technology, Hong Kong - This email address is being protected from spambots. You need JavaScript enabled to view it.
Recently, many methods have been developed for solving partial differential equations (PDEs) by neural networks. However, the curse of dimensionality (CoD) is a serious issue that generally exists in this field when dealing with high dimensional problems. In this work, we establish a new method for the approximation of functionals by neural networks without CoD by defining (i) a Fourier-type series on the infinite-dimensional space of functionals and (ii) the associated Barron spectral space $\mathcal{B}_s$ and a Hilbert space $\mathcal{H}_s$ of functionals. The approximation error of the designed neural network in this method is $O(1/\sqrt{m})$ where $m$ is the size of networks. Then, the proposed method is employed in several numerical experiments, such as evaluating the energy functionals, solving two-dimensional and four-dimensional Poisson equations by aforementioned neural networks at one or a few given points.
Joint work with YANG Yahong (The Hong Kong University of Science and Technology, Hong Kong) and XIANG Yang (The Hong Kong University of Science and Technology, Hong Kong).