Search Results for author:
Found 14 papers, 2 papers with code
Large Language Models for Mathematicians
Large language models (LLMs) such as ChatGPT have received immense interest for their general-purpose language understanding and, in particular, their ability to generate high-quality text or computer code.
Optimal learning of high-dimensional classification problems using deep neural networks
We study the problem of learning classification functions from noiseless training samples, under the assumption that the decision boundary is of a certain regularity.
Deep Microlocal Reconstruction for Limited-Angle Tomography
We present a deep learning-based algorithm to jointly solve a reconstruction problem and a wavefront set extraction problem in tomographic imaging.
The Modern Mathematics of Deep Learning
We describe the new field of mathematical analysis of deep learning.
Neural network approximation and estimation of classifiers with classification boundary in a Barron class
We prove bounds for the approximation and estimation of certain binary classification functions using ReLU neural networks.
Numerical Solution of the Parametric Diffusion Equation by Deep Neural Networks
Here, approximation theory predicts that the performance of the model should depend only very mildly on the dimension of the parameter space and is determined by the intrinsic dimension of the solution manifold of the parametric partial differential equation.
Approximation in $L^p(μ)$ with deep ReLU neural networks
In particular, the generalized results apply in the usual setting of statistical learning theory, where one is interested in approximation in $L^2(\mathbb{P})$, with the probability measure $\mathbb{P}$ describing the distribution of the data.
A Theoretical Analysis of Deep Neural Networks and Parametric PDEs
We derive upper bounds on the complexity of ReLU neural networks approximating the solution maps of parametric partial differential equations.
Error bounds for approximations with deep ReLU neural networks in $W^{s,p}$ norms
We analyze approximation rates of deep ReLU neural networks for Sobolev-regular functions with respect to weaker Sobolev norms.
Extraction of digital wavefront sets using applied harmonic analysis and deep neural networks
Microlocal analysis provides deep insight into singularity structures and is often crucial for solving inverse problems, predominately, in imaging sciences.
Equivalence of approximation by convolutional neural networks and fully-connected networks
Convolutional neural networks are the most widely used type of neural networks in applications.
Topological properties of the set of functions generated by neural networks of fixed size
We analyze the topological properties of the set of functions that can be implemented by neural networks of a fixed size.
General Topology Functional Analysis 54H99, 68T05, 52A30
Optimal approximation of piecewise smooth functions using deep ReLU neural networks
We study the necessary and sufficient complexity of ReLU neural networks---in terms of depth and number of weights---which is required for approximating classifier functions in $L^2$.
Optimal Approximation with Sparsely Connected Deep Neural Networks
Specifically, all function classes that are optimally approximated by a general class of representation systems---so-called \emph{affine systems}---can be approximated by deep neural networks with minimal connectivity and memory requirements.
