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math.NA2026
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…
math.NA2025
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…
math.NA2025
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…