Abstract

Scientific machine learning provides new tools for solving partial differential equations, but reliable performance often requires incorporating mathematical and numerical structures into the learning process. In this talk, I will first discuss numerics-guided scientific machine learning methods for challenging PDEs, which combine neural network-based coefficient learning with classical numerical trial spaces, drawing on finite element, discontinuous Galerkin, spectral element, and Müntz-type element methods. By exploiting problem-adapted approximation spaces and stability mechanisms, these methods are effective for PDEs with fluid flow, multiscale behavior, discontinuities, or singular perturbations. I will then turn to the theory behind such methods, presenting results on the approximation power of deep neural networks, including dimension-independent rates for Barron-type and logarithmic Barron classes. Finally, I will discuss how these theoretical tools extend to modern foundation models, providing error bounds that help explain their accuracy and generalization. Throughout, I will emphasize how approximation theory and numerical analysis offer a unifying lens for understanding learning-based methods.

WebEx Link

https://nationaltaiwanuniversity-ksz.my.webex.com/nationaltaiwanuniversity-ksz.my/j.php?MTID=m5d027e9fc7e8ad15e510284567d2ecbe

Organizer

Te-Sheng Lin (NYCU)