Related papers: Non-Linear Interactions in Neural Network Operator…
In this paper, we investigate the approximation behavior of both one and multidimensional neural network type operators for functions in $L^p(I^d,\rho)$, where $1\leq p<\infty$, associated with a general measure $\rho$ defined over a…
Nonlinear dynamical systems with symmetries exhibit a rich variety of behaviors, including complex attractor-basin portraits and enhanced and suppressed bifurcations. Symmetry arguments provide a way to study these collective behaviors and…
We provide a novel transcription of monotone operator theory to the non-Euclidean finite-dimensional spaces $\ell_1$ and $\ell_{\infty}$. We first establish properties of mappings which are monotone with respect to the non-Euclidean norms…
This is an update on the quasicentral modulus, an invariant for an n-tuple of Hilbert space operators and a rearrangement invariant norm, that plays a key-role in sharp multivariable generalizations of the classical Weyl-von Neumann-Kuroda…
We integrate neural operators with diffusion models to address the spectral limitations of neural operators in surrogate modeling of turbulent flows. While neural operators offer computational efficiency, they exhibit deficiencies in…
This work establishes the first rigorous stability guarantees for approximate predictors in delay-adaptive control of nonlinear systems, addressing a key challenge in practical implementations where exact predictors are unavailable. We…
This work focuses on developing methods for approximating the solution operators of a class of parametric partial differential equations via neural operators. Neural operators have several challenges, including the issue of generating…
The classical development of neural networks has primarily focused on learning mappings between finite dimensional Euclidean spaces or finite sets. We propose a generalization of neural networks to learn operators, termed neural operators,…
This paper introduces significant advancements in fractional neural operators (FNOs) through the integration of adaptive hybrid kernels and stochastic multiscale analysis. We address several open problems in the existing literature by…
In this paper, we provide two algorithms based on the theory of multidimensional neural network (NN) operators activated by hyperbolic tangent sigmoidal functions. Theoretical results are recalled to justify the performance of the here…
The basic results for nonlinear operators are given. These results include nonlinear versions of classical uniform boundedness theorem and Hahn-Banach theorem. Furthermore, the mappings from a metrizable space into another normed space can…
We present a data-driven pipeline for model building that combines interpretable machine learning, hydrodynamic theories, and microscopic models. The goal is to uncover the underlying processes governing nonlinear dynamics experiments. We…
In this paper we present a systematic study of regular sequences of quasi-nonexpansive operators in Hilbert space. We are interested, in particular, in weakly, boundedly and linearly regular sequences of operators. We show that the type of…
We study the approximation of functions which are invariant with respect to certain permutations of the input indices using flow maps of dynamical systems. Such invariant functions includes the much studied translation-invariant ones…
We consider the following learning problem: Given sample pairs of input and output signals generated by an unknown nonlinear system (which is not assumed to be causal or time-invariant), we wish to find a continuous-time recurrent neural…
Skewing operators play a central role in the symmetric function theory because of the importance of the product structure of the symmetric function space. The theory of noncommutative symmetric functions is a useful tool for studying…
Convolutional Neural Networks (CNNs) have been widely applied. But as the CNNs grow, the number of arithmetic operations and memory footprint also increase. Furthermore, typical non-linear activation functions do not allow associativity of…
After carrying out an overview on the non Euclidean geometrical setting suitable for the study of Kolmogorov operators with rough coefficients, we list some properties of the functional space $\mathcal{W}$, mirroring the classical $H^1$…
Hyperbolic geometry has emerged as a powerful tool for modeling complex, structured data, particularly where hierarchical or tree-like relationships are present. By enabling embeddings with lower distortion, hyperbolic neural networks offer…
Deep neural networks have emerged as powerful tools for learning operators defined over infinite-dimensional function spaces. However, existing theories frequently encounter difficulties related to dimensionality and limited…