报告题目：Solving multiscale elliptic problems by sparse radial basis function neural networks

报告摘要：

(1) In this work, we propose a sparse radial basis function neural network method to solve elliptic partial differential equations (PDEs) with multiscale coefficients. Inspired by the deep mixed residual method, we rewrite the second-order problem into a first-order system and employ multiple radial basis function neural networks (RBFNNs) to approximate unknown functions in the system. To avoid the overfitting due to the simplicity of RBFNN, an additional regularization is introduced and the loss function contains two parts: the $L_2$ loss for the residual of the first-order system and boundary conditions, and the $\ell_1$ regularization term for the weights of radial basis functions (RBFs).

（2）The accuracy and effectiveness of the proposed method are demonstrated through a collection of multiscale problems with scale separation, discontinuity and multiple scales from one to three dimensions. Notably, the $\ell_1$ regularization can achieve the goal of representing the solution by fewer RBFs. As a consequence, the total number of RBFs scales like $\mathcal{O}(\varepsilon^{-n\tau})$, where $\varepsilon$ is the smallest scale, $n$ is the dimensionality, and $\tau$ is typically smaller than $1$. It is worth mentioning that the proposed method not only has the numerical convergence and thus provides a reliable numerical solution in three dimensions when a classical method is typically not affordable, but also outperforms most other available machine leaning methods in terms of accuracy and robustness.

报告时间：2023.05.13   下午16:00-19:00

报告形式：腾讯会议；  会议号：898-226-442

获取会议密码请发邮件至：yangchang@hit.edu.cn

   报告人简介：陈景润，中国科学技术大学教授，主要从事材料性质的多尺度建模、分析与计算，以及机器学习与微分方程数值解。主要工作发表在J. Comput. Phys.，Math. Comp.，SIAM系列期刊等期刊上。