相关论文: Spectral and Nilpotent Matrix Orderings: Compariso…

Matrix inequalities play a pivotal role in mathematics, generalizing scalar inequalities and providing insights into linear operator structures. However, the widely used L\"owner ordering, which relies on real-valued eigenvalues, is limited…

A novel State-Space Neural Network with Ordered variance (SSNNO) is presented in which the state variables are ordered in decreasing variance. A systematic way of model order reduction with SSNNO is proposed, which leads to a Reduced order…

Matrix functions extend scalar function concepts to linear operators, offering a unified framework with broad applications in mathematics, science, and engineering. Classical definitions--via power series, spectral calculus, or Jordan…

We develop a unified matrix-spectral framework for analyzing stability and interpretability in deep neural networks. Representing networks as data-dependent products of linear operators reveals spectral quantities governing sensitivity to…

Partial differential equations (PDEs) govern complex systems, yet neural operators often struggle to efficiently capture the long-range, nonlocal interactions inherent in their solution maps. We introduce Spectral Filtering Operator (SFO),…

Stability certification and identifying a safe and stabilizing initial set are two important concerns in ensuring operational safety, stability, and robustness of dynamical systems. With the advent of machine-learning tools, these issues…

Accurate modeling of spatiotemporal dynamics is crucial to understanding complex phenomena across science and engineering. However, this task faces a fundamental challenge when the governing equations are unknown and observational data are…

Neural operators offer powerful approaches for solving parametric partial differential equations, but extending them to spherical domains remains challenging due to the need to preserve intrinsic geometry while avoiding distortions that…

This paper investigates the design of a robust fixed-order controller for single-input-single-output (SISO) polytopic systems with interval uncertainties, with the aim that the closed-loop stability is appropriately ensured and the…

Recently, the targeted ENO (TENO) schemes give a novel framework to keep optimal high-order spatial reconstruction wherever discontinuity is deemed to be vanished, including at smooth critical points, and to avoid oscillations by completely…

Spectral operators of matrices proposed recently in [C. Ding, D.F. Sun, J. Sun, and K.C. Toh, Math. Program. {\bf 168}, 509--531 (2018)] are a class of matrix valued functions, which map matrices to matrices by applying a vector-to-vector…

This paper explores operators with countable, continuous, and hybrid spectra, focusing on both finite dimensional and infinite dimensional cases, particularly in non-Hermitian systems. For finite dimensional operators, a novel concept of…

A nonzero pattern is a matrix with entries in {0,*}. A pattern is potentially nilpotent if there is some nilpotent real matrix with nonzero entries in precisely the entries indicated by the pattern. We develop ways to construct some…

Neural networks have become increasingly popular in controller design due to their versatility and efficiency. However, their integration into feedback systems can pose stability challenges, particularly in the presence of uncertainties.…

Goodness-of-fit tests are crucial tools for assessing the validity of statistical models. In this paper, we introduce a novel approach, the Spectral Smooth Test (SST), that generalizes Neyman's smooth test to high-dimensional data settings.…

We describe a seriation algorithm for ranking a set of items given pairwise comparisons between these items. Intuitively, the algorithm assigns similar rankings to items that compare similarly with all others. It does so by constructing a…

A plentitude of applications in scientific computing requires the approximation of mappings between Banach spaces. Recently introduced Fourier Neural Operator (FNO) and Deep Operator Network (DeepONet) can provide this functionality. For…

Learning maps between function spaces with a strong inductive bias is a central challenge in soft computing, especially when training data are scarce and standard deep architectures overfit. We introduce a \emph{neural integral operator}…

Scaling large models requires optimization strategies that ensure rapid convergence grounded in stability. Maximal Update Parametrization ($\boldsymbol{\mu}$P) provides a theoretical safeguard for width-invariant $\Theta(1)$ activation…

Several applications of spectral methods to problems related to the relativistic astrophysics of compact objects are presented. Based on a proper definition of the analytical properties of regular tensorial functions we have developed a…