34 citations · 212 across the 40 of their papers we have counts for
42 papers
Feature Learning for the High Dimensional Stationary Schödinger Equation with Deep Ritz Method
Yao Yao, Yulong Lu, Gilad Lerman
This paper investigates feature learning within the framework of the deep Ritz method for solving the stationary Schrödinger equation with Neumann boundary conditions. We first ana…
Posterior Concentration of Bayesian Physics-Informed Neural Networks for Elliptic PDEs
Yuxuan Zhao, Yulong Lu
We study the posterior contraction rate of Bayesian Physics-Informed Neural Networks (PINNs) for solving a general class of elliptic partial differential equations (PDEs). We focus…
A Theory of Diversity for Random Matrices with Applications to In-Context Learning of Schrödinger Equations
Frank Cole, Yulong Lu, Shaurya Sehgal
We address the following question: given a collection of independent random matrices drawn from a common distribution $…
In-Context Operator Learning on the Space of Probability Measures
Frank Cole, Dixi Wang, Yineng Chen +2
We introduce \emph{in-context operator learning on probability measure spaces} for optimal transport (OT). The goal is to learn a single solution operator that maps a pair of distr…
On the Dimension-Free Approximation of Deep Neural Networks for Symmetric Korobov Functions
Yulong Lu, Tong Mao, Jinchao Xu +1
Deep neural networks have been widely used as universal approximators for functions with inherent physical structures, including permutation symmetry. In this paper, we construct s…
Improving Data Fidelity via Diffusion Model-based Correction and Super-Resolution
Wuzhe Xu, Yulong Lu, Sifan Wang +1
We propose a unified diffusion model-based correction and super-resolution method to enhance the fidelity and resolution of diverse low-quality data through a two-step pipeline. Fi…
Convergence of Time-Averaged Mean Field Gradient Descent Dynamics for Continuous Multi-Player Zero-Sum Games
Yulong Lu, Pierre Monmarché
The approximation of mixed Nash equilibria (MNE) for zero-sum games with mean-field interacting players has recently raised much interest in machine learning. In this paper we prop…
Diffusion-based Models for Unpaired Super-resolution in Fluid Dynamics
Wuzhe Xu, Yulong Lu, Lian Shen +2
High-fidelity, high-resolution numerical simulations are crucial for studying complex multiscale phenomena in fluid dynamics, such as turbulent flows and ocean waves. However, dire…
In-Context Learning of Linear Dynamical Systems with Transformers: Approximation Bounds and Depth-Separation
Frank Cole, Yuxuan Zhao, Yulong Lu +1
This paper investigates approximation-theoretic aspects of the in-context learning capability of the transformers in representing a family of noisy linear dynamical systems. Our fi…
Wall laws for viscous flows in 3D randomly rough pipes: optimal convergence rates and stochastic integrability
Mitsuo Higaki, Yulong Lu, Jinping Zhuge
This paper is concerned with effective approximations and wall laws of viscous laminar flows in 3D pipes with randomly rough boundaries. The random roughness is characterized by th…
In-Context Learning of Linear Systems: Generalization Theory and Applications to Operator Learning
Frank Cole, Yulong Lu, Wuzhe Xu +1
We study theoretical guarantees for solving linear systems in-context using a linear transformer architecture. For in-domain generalization, we provide neural scaling laws that bou…
Generative downscaling of PDE solvers with physics-guided diffusion models
Yulong Lu, Wuzhe Xu
Solving partial differential equations (PDEs) on fine spatio-temporal scales for high-fidelity solutions is critical for numerous scientific breakthroughs. Yet, this process can be…
Fully discretized Sobolev gradient flow for the Gross-Pitaevskii eigenvalue problem
Ziang Chen, Jianfeng Lu, Yulong Lu +1
This paper studies the numerical approximation of the ground state of the Gross-Pitaevskii (GP) eigenvalue problem with a fully discretized Sobolev gradient flow induced by the $H^…
Score-based generative models break the curse of dimensionality in learning a family of sub-Gaussian probability distributions
Frank Cole, Yulong Lu
While score-based generative models (SGMs) have achieved remarkable success in enormous image generation tasks, their mathematical foundations are still limited. In this paper, we…
Optimal Deep Neural Network Approximation for Korobov Functions with respect to Sobolev Norms
Yahong Yang, Yulong Lu
This paper establishes the nearly optimal rate of approximation for deep neural networks (DNNs) when applied to Korobov functions, effectively overcoming the curse of dimensionalit…
Two-Scale Gradient Descent Ascent Dynamics Finds Mixed Nash Equilibria of Continuous Games: A Mean-Field Perspective
Yulong Lu
Finding the mixed Nash equilibria (MNE) of a two-player zero sum continuous game is an important and challenging problem in machine learning. A canonical algorithm to finding the M…
On the convergence of Sobolev gradient flow for the Gross-Pitaevskii eigenvalue problem
Ziang Chen, Jianfeng Lu, Yulong Lu +1
We study the convergences of three projected Sobolev gradient flows to the ground state of the Gross-Pitaevskii eigenvalue problem. They are constructed as the gradient flows of th…
Transfer Learning Enhanced DeepONet for Long-Time Prediction of Evolution Equations
Wuzhe Xu, Yulong Lu, Li Wang
Deep operator network (DeepONet) has demonstrated great success in various learning tasks, including learning solution operators of partial differential equations. In particular, i…
Birth-death dynamics for sampling: Global convergence, approximations and their asymptotics
Yulong Lu, Dejan Slepčev, Lihan Wang
Motivated by the challenge of sampling Gibbs measures with nonconvex potentials, we study a continuum birth-death dynamics. We improve results in previous works [51,57] and provide…
A Regularity Theory for Static Schrödinger Equations on in Spectral Barron Spaces
Ziang Chen, Jianfeng Lu, Yulong Lu +1
Spectral Barron spaces have received considerable interest recently as it is the natural function space for approximation theory of two-layer neural networks with a dimension-free…
Solving multiscale steady radiative transfer equation using neural networks with uniform stability
Yulong Lu, Li Wang, Wuzhe Xu
This paper concerns solving the steady radiative transfer equation with diffusive scaling, using the physics informed neural networks (PINNs). The idea of PINNs is to minimize a le…
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…
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…