相关论文: Universal Approximation of Nonlinear Operators and…

We obtain (i) a new, coordinate free, characterization of quasidiagonal operators with essential spectra contained in the unit circle by adapting the proof of a classical result in the theory of Banach spaces, (ii) an affirmative answer to…

Neural operator architectures employ neural networks to approximate operators mapping between Banach spaces of functions; they may be used to accelerate model evaluations via emulation, or to discover models from data. Consequently, the…

In this paper, we explain the universal approximation capabilities of deep residual neural networks through geometric nonlinear control. Inspired by recent work establishing links between residual networks and control systems, we provide a…

Shape optimization under uncertainty (OUU) is computationally intensive for classical PDE-based methods due to the high cost of repeated sampling-based risk evaluation across many uncertainty realizations and varying geometries, while…

We investigate spaceability phenomena in linear dynamics from a structural perspective. Given a continuous linear operator \(T:X \to X\), we introduce the set \(\Omega(T)\), consisting of all continuous linear operators \(h:X \to X\) for…

Neural networks are increasingly evolving towards training large models with big data, a method that has demonstrated superior performance across many tasks. However, this approach introduces an urgent problem: current deep learning models…

A bounded linear operator $U$ between Banach spaces is universal for the complement of some operator ideal $\mathfrak{J}$ if it is a member of the complement and it factors through every element of the complement of $\mathfrak{J}$. In the…

In Artificial Intelligence (AI) and computational science, learning the mappings between functions (called operators) defined on complex computational domains is a common theoretical challenge. Recently, Neural Operator emerged as a…

Uncertainty-aware deep learning (DL) models recently gained attention in fault diagnosis as a way to promote the reliable detection of faults when out-of-distribution (OOD) data arise from unseen faults (epistemic uncertainty) or the…

We will give an outline of the main results in our recent AMS Memoir, and include some new results, exposition and open problems. In that memoir we developed a general dilation theory for operator valued measures acting on Banach spaces…

The study of universal approximation of arbitrary functions $f: \mathcal{X} \to \mathcal{Y}$ by neural networks has a rich and thorough history dating back to Kolmogorov (1957). In the case of learning finite dimensional maps, many authors…

We introduce SILAS, a data-driven framework for discovering polynomial ordinary differential equations (ODEs) with provably bounded trajectories. Boundedness is certified by compact absorbing sets defined via polynomial Lyapunov functions.…

Let $\mathcal{L}(X;Y)$ be the space of bounded linear operators from a Banach space $X$ to a Banach space $Y$. Given an operator-valued function $u:\mathbb{R}_{\geq 0}\rightarrow \mathcal{L}(X;Y)$, suppose that every orbit $t\mapsto u(t)x$…

Motivated by the developing mathematics of deep learning, we build universal functions approximators of continuous maps between arbitrary Polish metric spaces $\mathcal{X}$ and $\mathcal{Y}$ using elementary functions between Euclidean…

Unbalanced Optimal Transport (UOT) has emerged as a robust relaxation of standard Optimal Transport, particularly effective for handling outliers and mass variations. However, scalable algorithms for UOT, specifically those based on…

Deep Operator Networks (DeepONets) and their physics-informed variants have shown significant promise in learning mappings between function spaces of partial differential equations, enhancing the generalization of traditional neural…

Neural operators serve as universal approximators for general continuous operators. In this paper, we derive the approximation rate of solution operators for the nonlinear parabolic partial differential equations (PDEs), contributing to the…

In this paper, using the concept of unbounded absolute weak convergence ($uaw$-convergence, for short) in a Banach lattice, we define two classes of continuous operators, named $uaw$-Dunford-Pettis and $uaw$-compact operators. We…

We develop a noncommutative invariant theory for ordinary linear differential operators on Riemann surfaces. For a monic binomially normalized operator $L=\sum_{k=0}^n {n\choose k}a_kD^{\,n-k}$, $a_0=1$, with coefficients in an associative…

We study families of strongly elliptic, second order differential operators with singular coefficients on domains with conical points. We obtain uniform estimates on their inverses and on the regularity of the solutions to the associated…