196 citations · 1.6k across the 143 of their papers we have counts for
6 papers · 2 filters
Machine Learning For Elliptic PDEs: Fast Rate Generalization Bound, Neural Scaling Law and Minimax Optimality
Yiping Lu, Haoxuan Chen, Jianfeng Lu +2
In this paper, we study the statistical limits of deep learning techniques for solving elliptic partial differential equations (PDEs) from random samples using the Deep Ritz Method…
Low-rank approximation for multiscale PDEs
Ke Chen, Shi Chen, Qin Li +2
Historically, analysis for multiscale PDEs is largely unified while numerical schemes tend to be equation-specific. In this paper, we propose a unified framework for computing mult…
On the Representation of Solutions to Elliptic PDEs in Barron Spaces
Ziang Chen, Jianfeng Lu, Yulong Lu
Numerical solutions to high-dimensional partial differential equations (PDEs) based on neural networks have seen exciting developments. This paper derives complexity estimates of t…
A Priori Generalization Error Analysis of Two-Layer Neural Networks for Solving High Dimensional Schrödinger Eigenvalue Problems
Jianfeng Lu, Yulong Lu
This paper analyzes the generalization error of two-layer neural networks for computing the ground state of the Schrödinger operator on a -dimensional hypercube. We prove that t…
Structure-preserving numerical schemes for Lindblad equations
Yu Cao, Jianfeng Lu
We study a family of structure-preserving deterministic numerical schemes for Lindblad equations. This family of schemes has a simple form and can systemically achieve arbitrary hi…
A Priori Generalization Analysis of the Deep Ritz Method for Solving High Dimensional Elliptic Equations
Jianfeng Lu, Yulong Lu, Min Wang
This paper concerns the a priori generalization analysis of the Deep Ritz Method (DRM) [W. E and B. Yu, 2017], a popular neural-network-based method for solving high dimensional pa…