5 papers
Convergence of Least Squares Methods for PDEs on R^d with Shallow ReLU^k Networks
Tong Mao, Lei Shi, Hanfei Zhou
We establish inverse inequalities in Gaussian Sobolev norms for shallow trial spaces. The estimates identify the sharp regularity threshold and quantify the depen…
Simultaneous CNN Approximation on Manifolds with Applications to Boundary Value Problems
Hanfei Zhou, Lei Shi
This paper develops convolutional neural network (CNN) methods for simultaneous Sobolev approximation and elliptic boundary value problems on compact Riemannian manifolds. We prove…
A Quantitative Approximation Framework for Flow Distillation in Diffusion Models
Weiguo Gao, Ming Li, Lei Shi +1
We develop a quantitative framework for diffusion distillation by viewing few step sampling as approximation through compositions of learned flow maps. For trajectory distillation…
Expressive Power of Deep Networks on Manifolds: Simultaneous Approximation
Hanfei Zhou, Lei Shi
A key challenge in scientific machine learning is solving partial differential equations (PDEs) on complex domains, where the curved geometry complicates the approximation of funct…
Weak Physics Informed Neural Networks for Geometry Compatible Hyperbolic Conservation Laws on Manifolds
Hanfei Zhou, Lei Shi
Physics-informed neural networks (PINNs) provide a mesh-free approach to solving high-dimensional PDEs on complex geometries, but their theoretical foundations on manifolds remain…