Understanding and using DNN-based algorithm in solving PDE


主讲人:许志钦  上海交通大学副教授




内容介绍:We demonstrate a very universal Frequency Principle (F-Principle) --- DNNs often  fit target functions from low to high frequencies --- on high-dimensional  benchmark datasets and deep neural networks. We use F-Principle to understand  the difference of DNN with traditional methods. We then propose novel  multi-scale DNNs (MscaleDNN) using the idea of radial scaling in frequency  domain and activation functions with compact support. The radial scaling  converts the problem of approximation of high frequency content of the PDEs  solution to one of lower frequency, and the compact support activation functions  facilitate the separation of scales to be approximated by corresponding DNNs. As  a result, the MscaleDNNs achieve fast uniform convergence over multiple scales.  The proposed MscaleDNNs are shown to be superior to traditional fully connected  DNNs and can be used as an effective mesh-less numerical method for elliptic  PDEs.