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.
arXiv:2606. 04031v1 Announce Type: new Abstract: Coupled gradient descent--where the update of one parameter block depends on another--underlies bilevel optimization, two-time-scale stochastic approximation, and adversarial training.
The paper studies operator learning on function spaces using encoder–decoder architectures. It shows that as input and output resolutions grow, the induced kernels converge to a limiting kernel, enabling regularity assumptions independent of resolution. The authors derive upper and lower bounds for regularized stochastic gradient descent, extend the analysis to neural networks via the limiting neural tangent kernel, and provide error bounds and complexity guarantees for various kernel and encoding constructions.
arXiv:2607. 27781v1 Announce Type: cross Abstract: We establish a dimension-efficient neural network approximation theory for solutions to fractional parabolic equations with lower-order drift and potential terms.
arXiv:2608. 15982v1 Announce Type: new Abstract: We develop operator-theoretic generalization bounds for deep multi-output function classes by representing network layers as Koopman composition operators on vector-valued reproducing kernel Hilbert spaces.
arXiv:2609.15355v1 Announce Type: cross Abstract: We study the uniform approximation of smooth scalar-valued functionals on an infinite-dimensional separable Hilbert space by deep ReLU neural network...
arXiv:2605. 07116v2 Announce Type: replace Abstract: We analyze a neural semi-discrete method for high-dimensional first-order Hamilton-Jacobi-Bellman (HJB) equations with known or learned dynamics.
We develop operator-theoretic generalization bounds for deep multi-output function classes by representing network layers as Koopman composition operators on vector-valued reproducing kernel Hilbert spaces. In vector-valued Sobolev RKHSs, we derive Rademacher complexity bounds for invertible and width-expanding injective architectures.
arXiv:2608. 11479v1 Announce Type: new Abstract: We establish convergence guarantees of gradient descent for general feedforward neural networks of arbitrary width or depth, with no special requirements on the initialization or dataset.
arXiv:2507. 05164v2 Announce Type: replace-cross Abstract: In this chapter, we utilize dynamical systems to analyze several aspects of machine learning algorithms.
arXiv:2606. 08799v1 Announce Type: cross Abstract: We study the generalization of ridge-regularized nonlinear least-squares models via on-average algorithmic stability, deriving error bounds for local minimizers in terms of a data-dependent effective dimension that reflects the geometry of the gradient model at the trained parameters, through the empirical Jacobian Gram matrix and a residual--curvature term.
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.
arXiv:2607. 07468v1 Announce Type: cross Abstract: We study the recovery of sparse functions from finite, noisy, and indirect observations in the framework of statistical inverse learning.