11 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 an…
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…
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…
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…