Applied Mathematics Seminar——Solving Nonlinear PDEs with Sparse Radial Basis Function Networks
报告人：Xiaochuan Tian (University of California, San Diego)
时间：2025-07-02 10:00-11:00
地点：智华楼-四元厅225
Abstract:
We propose a general framework for solving nonlinear PDEs using neural networks. To avoid over-parameterization and eliminate redundant features, a regularization approach that encourages sparsity is explored within the framework of shallow radial basis function (RBF) networks.
An adaptive training process iteratively adds neurons to maintain a compact network structure.
Existence theory is established via the calculus of variations, and a representer theorem is derived for reproducing kernel Banach spaces (RKBS) associated with one-hidden-layer neural networks of possibly infinite width. An error estimate for the neural network approximation to the PDE is also derived. Training is performed using a second-order semismooth Newton method with gradient boosting. The approach is compared with the reproducing kernel Hilbert space (RKHS) framework and Gaussian process methods.
Bio:
Xiaochuan Tian is an Associate Professor in the Department of Mathematics at the University of California, San Diego. Her research focuses on mathematical modeling, applied analysis, and numerical methods for partial differential and integro-differential equations. Her current interests include nonlocal modeling, high-order and meshfree numerical schemes, asymptotically compatible and multiscale methods, as well as the emerging interface between machine learning and partial differential equations.
