Related papers: Geometric Kolmogorov-Arnold Superposition Theorem
A key problem of solid mechanics is the identification of the constitutive law of a material, that is, the relation between strain and stress. Machine learning has lead to considerable advances in this field lately. Here we introduce…
We show that for any finite-dimensional quantum systems the conserved quantities can be characterized by their robustness to small perturbations: for fragile symmetries small perturbations can lead to large deviations over long times, while…
In this paper, we gave a weighted compactness theory for the generalized commutators of vecotor-valued multilinear Calder\'{o}n-Zygmund operators. This was done by establishing a weighted Fr\'{e}chet-Kolmogorov theorem, which holds for…
Compatible equations, Singularities of solutions, Topological charges and quasi-charges. PhD thesis (translated frorm Russian). The book shows the sights of Absolute Parallelism (AP), and contains useful information on the problem of…
This paper examines and strengthens the Cuntz-Thomsen picture of equivariant Kasparov theory for arbitrary second-countable locally compact groups, in which elements are given by certain pairs of cocycle representations between C*-dynamical…
Non-linear partial differential Kolmogorov equations are successfully used to describe a wide range of time dependent phenomena, in natural sciences, engineering or even finance. For example, in physical systems, the Allen-Cahn equation…
Tabular data is the most common type of data in real-life scenarios. In this study, we propose the TabKANet model for tabular data modeling, which targets the bottlenecks in learning from numerical content. We constructed a…
This paper focuses on the minimization of a sum of a twice continuously differentiable function $f$ and a nonsmooth convex function. An inexact regularized proximal Newton method is proposed by an approximation to the Hessian of $f$…
A compact subset $K$ of the complex plane $\C$ is a set of polynomial (respectively rational) approximation if $P(K)=A(K)$ (respectively $R(K)=A(K)$), where $P(K)$ (respectively $R(K)$) is the family of functions on $K$ which are uniform…
The domain of laser fusion presents a unique and challenging predictive modeling application landscape for machine learning methods due to high problem complexity and limited training data. Data-driven approaches utilizing prescribed…
This paper presents the use of Kolmogorov-Arnold Networks (KANs) for forecasting the CBOE Volatility Index (VIX). Unlike traditional MLP-based neural networks that are often criticized for their black-box nature, KAN offers an interpretable…
We take the first step in the development of an equivariant version of modern, Gromov-style Oka theory. We define equivariant versions of the standard Oka property, ellipticity, and homotopy Runge property of complex manifolds, show that…
The Stone-von Neumann Theorem is a fundamental result which unified the competing quantum mechanical models of matrix mechanics and wave mechanics. It's mechanism of proof ultimately involved the study of unitary group representations on a…
In the setting of CAT(k) spaces, common fixed point iterations built from prox mappings (e.g. prox-prox, Krasnoselsky-Mann relaxations, nonlinear projected-gradients) converge locally linearly under the assumption of linear metric…
We develop an abstract KAM theorem for systems of infinitely many interacting particles with decaying masses and all-to-all interactions. Using this framework, we construct full-dimensional KAM tori for infinite-dimensional mechanical…
Equivariant $K$-theory is a generalized equivariant cohomology theory which is a hybrid of the $K$-theory of a topological space and the representation theory of the group acting on it. In this article, we review the basics of equivariant…
Quantum state tomography (QST), the process of reconstructing some unknown quantum state $\hat\rho$ from repeated measurements on copies of said state, is a foundationally important task in the context of quantum computation and simulation.…
Assembly Theory (AT) and its central measure, the assembly index (Ai), represent an invaluable opportunity to address some of the most persistent and widespread conflations and misconceptions about computability and complexity theory in…
Kolmogorov decomposition for a given completely positive definite kernel is a generalization of Paschke's GNS construction for the completely positive map. Using Kolmogorov decomposition, to every quantum dynamical semigroup (QDS) for…
Reconstructing time-resolved flow fields from temporally sparse velocimetry measurements is critical for characterizing many complex thermal-fluid systems. We introduce a machine learning framework for uncertainty-aware flow reconstruction…