arXiv Machine Learning July 3, 2026 By Matej Benko, Pierre Bousquet, Iwona Chlebicka, B{\l}a\.zej Miasojedow

Born Discrete, Made Smooth: Variational Formulation of Shallow Neural Networks

Read the original on arXiv Machine Learning →

arXiv:2607. 02003v1 Announce Type: cross Abstract: Although neural networks are remarkably effective, their underlying optimization principles remain theoretically elusive, often characterized by non-convex landscapes and stochastic heuristics.

Summary generated by The Flow from the publisher's feed. The full article lives at arXiv Machine Learning.

arXiv Machine Learning

Jul 23

A convergence result of a continuous model of deep learning via a \L{}ojasiewicz--Simon inequality

arXiv:2311. 15365v3 Announce Type: replace Abstract: We study an idealized training process for deep neural networks in a continuous-depth, mean-field model in which each layer is parameterized by a probability measure on a Euclidean parameter space.

By Noboru Isobe

arXiv Machine Learning

Jul 14

A Learning-Based Ansatz Satisfying Boundary Conditions in Variational Problems

arXiv:2505. 12430v2 Announce Type: replace Abstract: Recently, innovative adaptations of the Ritz method incorporating deep learning have been developed, known as the Deep Ritz Method.

By Rafael Florencio, Julio Guerrero

arXiv Machine Learning

Aug 11

Generalized Convexity and Smoothness via Conjugate Duality: Optimization Theory for Deep Neural Networks

arXiv:2608. 09523v1 Announce Type: new Abstract: Deep neural network (DNN) training with stochastic gradient descent (SGD) and its variants achieves strong empirical performance, yet classical optimization theory does not fully explain this success.

By Binchuan Qi

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arXiv Machine Learning

Jun 16

Representation Costs in Data Science: Foundations and the Quasi-Banach Spaces of Deep Neural Networks

arXiv:2606. 14954v1 Announce Type: cross Abstract: We develop a general framework for analyzing representation costs of parametric data-fitting methods through their parameter-space regularizers.

By Greg Ongie, Rahul Parhi

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arXiv Machine Learning

Jun 10

Deeper or Wider: A Perspective from Optimal Generalization Error with Sobolev Loss

arXiv:2402. 00152v5 Announce Type: replace Abstract: Constructing the architecture of a neural network is a challenging pursuit for the machine learning community, and the dilemma of whether to go deeper or wider remains a persistent question.

By Yahong Yang, Juncai He

arXiv Machine Learning

Jul 23

Non-Asymptotic Variational Learning for Monotone Nonlinear Multiscale Elliptic Equations: Scale-Robust Primal-Dual Bounds and Strong-Form Statistical Ill-Conditioning

arXiv:2607. 15702v2 Announce Type: replace-cross Abstract: We develop a non-asymptotic approximation, sampling, and finite-iteration optimization theory for variational physics-informed approximation of uniformly monotone nonlinear multiscale elliptic equations.

By Ronald Katende

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A convergence result of a continuous model of deep learning via a \L{}ojasiewicz--Simon inequality

A Learning-Based Ansatz Satisfying Boundary Conditions in Variational Problems

Generalized Convexity and Smoothness via Conjugate Duality: Optimization Theory for Deep Neural Networks

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