196 citations · 1.6k across the 143 of their papers we have counts for
50 papers · 1 filter
Score-based Transport Modeling for Mean-Field Fokker-Planck Equations
Jianfeng Lu, Yue Wu, Yang Xiang
We use the score-based transport modeling method to solve the mean-field Fokker-Planck equations, which we call MSBTM. We establish an upper bound on the time derivative of the Kul…
Computing Truncated Joint Approximate Eigenbases for Model Order Reduction
Terry A. Loring, Fredy Vides
In this document, some elements of the theory and algorithmics corresponding to the existence and computability of approximate joint eigenpairs for finite collections of matrices w…
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…
Manifold Learning and Nonlinear Homogenization
Shi Chen, Qin Li, Jianfeng Lu +1
We describe an efficient domain decomposition-based framework for nonlinear multiscale PDE problems. The framework is inspired by manifold learning techniques and exploits the tang…
Fast localization of eigenfunctions via smoothed potentials
Jianfeng Lu, Cody Murphey, Stefan Steinerberger
We study the problem of predicting highly localized low-lying eigenfunctions in bounded domains for rapidly varying potentials . Filoche &…
A Proximal-Gradient Algorithm for Crystal Surface Evolution
Katy Craig, Jian-Guo Liu, Jianfeng Lu +2
As a counterpoint to recent numerical methods for crystal surface evolution, which agree well with microscopic dynamics but suffer from significant stiffness that prevents simulati…
Numerical analysis for inchworm Monte Carlo method: Sign problem and error growth
Zhenning Cai, Jianfeng Lu, Siyao Yang
We consider the numerical analysis of the inchworm Monte Carlo method, which is proposed recently to tackle the numerical sign problem for open quantum systems. We focus on the gro…
Complexity of randomized algorithms for underdamped Langevin dynamics
Yu Cao, Jianfeng Lu, Lihan Wang
We establish an information complexity lower bound of randomized algorithms for simulating underdamped Langevin dynamics. More specifically, we prove that the worst strong er…
Ensemble Kalman Inversion for nonlinear problems: weights, consistency, and variance bounds
Zhiyan Ding, Qin Li, Jianfeng Lu
Ensemble Kalman Inversion (EnKI) and Ensemble Square Root Filter (EnSRF) are popular sampling methods for obtaining a target posterior distribution. They can be seem as one step (t…
Universal approximation of symmetric and anti-symmetric functions
Jiequn Han, Yingzhou Li, Lin Lin +3
We consider universal approximations of symmetric and anti-symmetric functions, which are important for applications in quantum physics, as well as other scientific and engineering…
Fisher information regularization schemes for Wasserstein gradient flows
Wuchen Li, Jianfeng Lu, Li Wang
We propose a variational scheme for computing Wasserstein gradient flows. The scheme builds upon the Jordan--Kinderlehrer--Otto framework with the Benamou-Brenier's dynamic formula…
A low-rank Schwarz method for radiative transport equation with heterogeneous scattering coefficient
Ke Chen, Qin Li, Jianfeng Lu +1
Random sampling has been used to find low-rank structure and to build fast direct solvers for multiscale partial differential equations of various types. In this work, we design an…
Tensor Ring Decomposition: Optimization Landscape and One-loop Convergence of Alternating Least Squares
Ziang Chen, Yingzhou Li, Jianfeng Lu
In this work, we study the tensor ring decomposition and its associated numerical algorithms. We establish a sharp transition of algorithmic difficulty of the optimization problem…
Variational training of neural network approximations of solution maps for physical models
Yingzhou Li, Jianfeng Lu, Anqi Mao
A novel solve-training framework is proposed to train neural network in representing low dimensional solution maps of physical models. Solve-training framework uses the neural netw…
Numerical methods for stochastic differential equations based on Gaussian mixture
Lei Li, Jianfeng Lu, Jonathan Mattingly +1
We develop in this work a numerical method for stochastic differential equations (SDEs) with weak second order accuracy based on Gaussian mixture. Unlike the conventional higher or…
A Numerical method for coupling the BGK model and Euler equation through the linearized Knudsen layer
Hongxu Chen, Qin Li, Jianfeng Lu
The Bhatnagar-Gross-Krook (BGK) model, a simplification of the Boltzmann equation, in the absence of boundary effect, converges to the Euler equations when the Knudsen number is sm…
Random Sampling and Efficient Algorithms for Multiscale PDEs
Ke Chen, Qin Li, Jianfeng Lu +1
We describe a numerical framework that uses random sampling to efficiently capture low-rank local solution spaces of multiscale PDE problems arising in domain decomposition. In con…
Integrated Tempering Enhanced Sampling Method as the Infinite Switching Limit of Simulated Tempering
Zhiyi You, Liying Li, Jianfeng Lu +1
Fast and accurate sampling method is in high demand, in order to bridge the large gaps between molecular dynamic simulations and experimental observations. Recently, integrated tem…
Coordinate-wise descent methods for leading eigenvalue problem
Yingzhou Li, Jianfeng Lu, Zhe Wang
Leading eigenvalue problems for large scale matrices arise in many applications. Coordinate-wise descent methods are considered in this work for such problems based on a reformulat…