Speaker: Xiaochuan Tian, UC San Diego
Title: "Sparse Radial Basis Function Networks for Solving Nonlinear PDEs"
Abstract:
Solving nonlinear partial differential equations (PDEs) remains a central challenge in scientific computing. Traditional numerical methods are backed by theoretical guarantees but often require problem-specific designs and struggle with the curse of dimensionality, while neural-network-based approaches provide flexibility yet face difficulties such as nonconvex optimization, over-parameterization, and limited interpretability. In this talk, I will present a sparse radial basis function (RBF) network framework that, on one hand, functions as an adaptive collocation PDE solver, and on the other hand, as a shallow neural network with efficient training procedures and greater interpretability. The key idea lies in extending classical RBF collocation by optimizing nonlinear features, with sparsity-promoting regularization that prevents over-parameterization and removes redundant features.
On the theoretical side, the method is built on Reproducing Kernel Banach Spaces (RKBS) induced by one-hidden-layer neural networks of possibly infinite width. We prove a representer theorem showing that the sparse optimization problem in the RKBS admits a finite solution and establishes error bounds that offer a foundation for generalizing classical numerical analysis. On the computational side, the method is implemented via a three-phase algorithm that combines adaptive feature selection, second-order optimization, and pruning of inactive neurons. I will illustrate the framework with numerical experiments, showing both its effectiveness and scenarios where it provides clear advantages over Gaussian Process and RKHS-based methods. This framework exemplifies a new pathway toward adaptive PDE solvers that combine rigorous analysis with efficient, learning-inspired algorithms.
Bio:
Xiaochuan Tian is an Associate Professor of Mathematics at UC San Diego. Her research spans mathematical modeling, applied analysis, and numerical methods for differential and nonlocal equations, with recent work at the interface of machine learning and PDEs. She received her Ph.D. from Columbia University in 2017, was an R.H. Bing Instructor at UT Austin, and is a recipient of the Sloan Research Fellowship, NSF CAREER Award, and other honors.
In person attendance at this seminar is only open to Columbia University affiliates. External guests are welcome to attend remotely. Please contact [email protected] if you need the Zoom link for this seminar.
