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06/09/2023
Group Equivariant Fourier Neural Operators for Partial Differential Equations
We consider solving partial differential equations (PDEs) with Fourier n...
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02/10/2023
Achieving acceleration despite very noisy gradients
We present a novel momentum-based first order optimization method (AGNES...
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09/02/2022
Optimal bump functions for shallow ReLU networks: Weight decay, depth separation and the curse of dimensionality
In this note, we study how neural networks with a single hidden layer an...
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03/25/2022
Qualitative neural network approximation over R and C: Elementary proofs for analytic and polynomial activation
In this article, we prove approximation theorems in classes of deep and ...
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06/04/2021
Stochastic gradient descent with noise of machine learning type. Part II: Continuous time analysis
The representation of functions by artificial neural networks depends on...
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05/04/2021
Stochastic gradient descent with noise of machine learning type. Part I: Discrete time analysis
Stochastic gradient descent (SGD) is one of the most popular algorithms ...
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12/10/2020
On the emergence of tetrahedral symmetry in the final and penultimate layers of neural network classifiers
A recent numerical study observed that neural network classifiers enjoy ...
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12/02/2020
Some observations on partial differential equations in Barron and multi-layer spaces
We use explicit representation formulas to show that solutions to certai...
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09/28/2020
A priori estimates for classification problems using neural networks
We consider binary and multi-class classification problems using hypothe...
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09/22/2020
Towards a Mathematical Understanding of Neural Network-Based Machine Learning: what we know and what we don't
The purpose of this article is to review the achievements made in the la...
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07/30/2020
On the Banach spaces associated with multi-layer ReLU networks: Function representation, approximation theory and gradient descent dynamics
We develop Banach spaces for ReLU neural networks of finite depth L and ...
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06/10/2020
Representation formulas and pointwise properties for Barron functions
We study the natural function space for infinitely wide two-layer neural...
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05/27/2020
On the Convergence of Gradient Descent Training for Two-layer ReLU-networks in the Mean Field Regime
We describe a necessary and sufficient condition for the convergence to ...
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05/21/2020
Can Shallow Neural Networks Beat the Curse of Dimensionality? A mean field training perspective
We prove that the gradient descent training of a two-layer neural networ...
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05/21/2020