arXiv:2112. 08125v3 Announce Type: replace-cross Abstract: We construct and analyze approximation rates of deep operator networks (ONets) between infinite-dimensional spaces that emulate with an exponential rate of convergence the coefficient-to-solution map of elliptic second-order partial differential equations.
By Carlo Marcati, Christoph Schwab
The paper investigates how parameter Jacobians influence the stability of network outputs within the framework of network dynamics, learning models, and neural tangent kernels (NTK). It demonstrates that linearized dynamics can be expressed as a semigroup of linear operators on Hilbert spaces, and provides explicit a priori perturbation bounds for fixed‑kernel linearizations in the NTK setting. The authors also offer refinements for task‑specific spaces, ergodic comparison estimates, spectral‑distribution conditions, and extensions to nonautonomous NTK evolutions, supported by worked examples.
By Halyun Jeong, Palle E. T. Jorgensen, Hyun-Kyoung Kwon, Myung-Sin Song, James Tian
arXiv:2607. 00320v1 Announce Type: cross Abstract: We establish approximation and learning guarantees for Fourier neural operators (FNOs) applied to time-$T$ solution operators of dissipative evolution equations.
By Nisha Chandramoorthy, Daniel Sanz-Alonso, Nathan Waniorek
arXiv:2602. 19381v2 Announce Type: replace-cross Abstract: We establish a regularity theorem for second-order elliptic PDEs on $\mathbb{R}^{d}$ in spectral Barron spaces.
By Ziang Chen, Liqiang Huang, Mengxuan Yang, Shengxuan 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: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.
By Matej Benko, Pierre Bousquet, Iwona Chlebicka, B{\l}a\.zej Miasojedow