50 citations · 193 across the 27 of their papers we have counts for
11 papers · 1 filter
On the emergence of simplex symmetry in the final and penultimate layers of neural network classifiers
Weinan E, Stephan Wojtowytsch
A recent numerical study observed that neural network classifiers enjoy a large degree of symmetry in the penultimate layer. Namely, if where is a linear map…
Some observations on high-dimensional partial differential equations with Barron data
Weinan E, Stephan Wojtowytsch
We use explicit representation formulas to show that solutions to certain partial differential equations lie in Barron spaces or multilayer spaces if the PDE data lie in such funct…
A priori estimates for classification problems using neural networks
Weinan E, Stephan Wojtowytsch
We consider binary and multi-class classification problems using hypothesis classes of neural networks. For a given hypothesis class, we use Rademacher complexity estimates and dir…
Towards a Mathematical Understanding of Neural Network-Based Machine Learning: what we know and what we don't
Weinan E, Chao Ma, Stephan Wojtowytsch +1
The purpose of this article is to review the achievements made in the last few years towards the understanding of the reasons behind the success and subtleties of neural network-ba…
On the Banach spaces associated with multi-layer ReLU networks: Function representation, approximation theory and gradient descent dynamics
Weinan E, Stephan Wojtowytsch
We develop Banach spaces for ReLU neural networks of finite depth and infinite width. The spaces contain all finite fully connected -layer networks and their -limiting…
Connected Coulomb Columns: Analysis and Numerics
Patrick Dondl, Matteo Novaga, Stephan Wojtowytsch +1
We consider a version of Gamow's liquid drop model with a short range attractive perimeter-penalizing potential and a long-range Coulomb interaction of a uniformly charged mass in…
Representation formulas and pointwise properties for Barron functions
Weinan E, Stephan Wojtowytsch
We study the natural function space for infinitely wide two-layer neural networks with ReLU activation (Barron space) and establish different representation formulae. In two cases,…
On the Convergence of Gradient Descent Training for Two-layer ReLU-networks in the Mean Field Regime
Stephan Wojtowytsch
We describe a necessary and sufficient condition for the convergence to minimum Bayes risk when training two-layer ReLU-networks by gradient descent in the mean field regime with o…
Can Shallow Neural Networks Beat the Curse of Dimensionality? A mean field training perspective
Stephan Wojtowytsch, Weinan E
We prove that the gradient descent training of a two-layer neural network on empirical or population risk may not decrease population risk at an order faster than un…
Kolmogorov Width Decay and Poor Approximators in Machine Learning: Shallow Neural Networks, Random Feature Models and Neural Tangent Kernels
Weinan E, Stephan Wojtowytsch
We establish a scale separation of Kolmogorov width type between subspaces of a given Banach space under the condition that a sequence of linear maps converges much faster on one o…
On the motion of curved dislocations in three dimensions: Simplified linearized elasticity
Irene Fonseca, Janusz Ginster, Stephan Wojtowytsch
It is shown that in core-radius cutoff regularized simplified elasticity (where the elastic energy depends quadratically on the full displacement gradient rather than its symmetriz…