This paper investigates the ρ^p-Lipschitz constants of deep ReLU neural networks with random weights drawn from a He‑style initialization. For zero‑bias networks, it provides high‑probability upper and lower bounds that differ by at most a logarithmic factor in depth, and shows a sharp contrast between the regimes p∈[1,2) and p∈[2,∞], with the former behaving like the Euclidean norm of a Gaussian vector and the latter like its dual norm. The analysis is extended to networks with non‑zero biases from symmetric distributions, yielding bounds that differ by a logarithmic factor in width and a linear factor in depth.
By Sjoerd Dirksen, Patrick Finke, Paul Geuchen, Dominik St\"oger, Felix Voigtlaender
arXiv:2607. 19167v1 Announce Type: cross Abstract: Motivated by the numerical computation of the Mean Escape Time (MET) $\tau:\Omega\to\mathbb{R}$ of a stochastic process from a bounded domain $\Omega\subseteq\mathbb{R}^d$, we study elliptic Dirichlet boundary value problems (BVPs) using boundary-enforced Physics-Informed Neural Networks (PINNs), in which the Dirichlet condition is imposed exactly by multiplying the network output with a predefined distance-to-boundary approximation $\rho$.
By Nathanael Tepakbong, Jun Fan, Xiang Zhou, Ding-Xuan Zhou
arXiv:2609. 03626v1 Announce Type: cross Abstract: Rigorous results show that feedforward neural networks can overcome the curse of dimensionality in the numerical approximation of high-dimensional partial differential equations (PDEs), but comparatively little is known about residual neural networks (ResNets) in the nonlinear PDE setting.
By Ilkhom Mukhammadiev, Diyora Salimova
arXiv:2605. 31152v2 Announce Type: replace-cross Abstract: This paper studies how efficiently deep ReLU neural networks can approximate and learn smooth functions.
By Yunfei Yang, Jun Fan
arXiv:2605. 08170v2 Announce Type: replace Abstract: Neural operators have emerged as a powerful tool for learning mappings between infinite-dimensional function spaces.
By Nicole Hao
arXiv:2607. 07778v1 Announce Type: new Abstract: Bubeck, Li and Nagaraj conjectured that, for generic data, any two-layer neural network with $m$ neurons that fits $n$ noisy labels must have Lipschitz constant at least of order $\sqrt{n/m}$, with no restriction on the size of the weights.
By Yitzchak Shmalo
An input may activate few hidden units even when different inputs collectively use an entire network. We study the statistical complexity of this input-dependent sparsity in the one-hidden-layer ReLU model of Awasthi et al.
arXiv:2605. 18528v2 Announce Type: replace-cross Abstract: A growing lesson from neural network optimization is that optimizer design should respect how the model is parametrized.
By Jiayu Zhang, Tianyi Lin
arXiv:2609. 05263v1 Announce Type: cross Abstract: We investigate the best $L_2$ approximation of mixed Sobolev spaces by shallow neural networks with $n$ neurons and general activation functions.
By Yuwen Li, Guozhi Zhang
arXiv:2606. 16975v1 Announce Type: cross Abstract: In this work, we investigate new activation functions for achieving arbitrary-accuracy Sobolev approximation by fixed-size neural networks.
By Baicheng Li, Haizhao Yang, Shijun Zhang
arXiv:2603. 28956v2 Announce Type: replace-cross Abstract: The minimum-norm interpolator (MNI) framework has recently attracted considerable attention as a tool for understanding generalization in overparameterized models, such as neural networks.
By Gil Kur, Pierre Bizeul
arXiv:2608. 06687v1 Announce Type: cross Abstract: We develop a rigorous theory of discrete residual least-squares approximation for elliptic spectral equations $\mathfrak L_\beta u=f$ using linearized ReLU$^k$ neural networks on the sphere, where $\mathfrak L_\beta$ is a positive elliptic spectral multiplier of order $\beta$.
By Xinliang Liu, Tong Mao, Jinchao Xu