Related papers: Spectral methods for Neural Integral Equations

IR or near-infrared (NIR) spectroscopy is a method used to identify a compound or to analyze the composition of a material. Calibration of NIR spectra refers to the use of the spectra as multivariate descriptors to predict concentrations of…

Solving partial differential equations (PDEs) by neural networks as well as Kolmogorov-Arnold Networks (KANs), including physics-informed neural networks (PINNs), physics-informed KANs (PIKANs), and neural operators, are known to exhibit…

Spectral methods have emerged as a simple yet surprisingly effective approach for extracting information from massive, noisy and incomplete data. In a nutshell, spectral methods refer to a collection of algorithms built upon the eigenvalues…

This article presents an argument for why quantum computers could unlock new methods for machine learning. We argue that spectral methods, in particular those that learn, regularise, or otherwise manipulate the Fourier spectrum of a machine…

Spectral problems governed by differential operators underpin a wide range of physical systems, yet remain computationally challenging because their spectra depend sensitively on continuous parameters and often demand repeated evaluations…

We describe a neural-based method for generating exact or approximate solutions to differential equations in the form of mathematical expressions. Unlike other neural methods, our system returns symbolic expressions that can be interpreted…

Nonlinear operators with long distance spatiotemporal dependencies are fundamental in modeling complex systems across sciences, yet learning these nonlocal operators remains challenging in machine learning. Integral equations (IEs), which…

We are concerned with the numerical solution of a class integro-differential equations, known as Neural Field Equations, which describe the large-scale dynamics of spatially structured networks of neurons. These equations have many…

Machine learning has emerged as an invaluable tool in many research areas. In the present work, we harness this power to predict highly accurate molecular infrared spectra with unprecedented computational efficiency. To account for…

Signal analysis and classification is fraught with high levels of noise and perturbation. Computer-vision-based deep learning models applied to spectrograms have proven useful in the field of signal classification and detection; however,…

We consider the problem of training input-output recurrent neural networks (RNN) for sequence labeling tasks. We propose a novel spectral approach for learning the network parameters. It is based on decomposition of the cross-moment tensor…

This work investigates two physics-based models that simulate the non-linear partial differential algebraic equations describing an electric double layer supercapacitor. In one model the linear dependence between electrolyte concentration…

Neural Networks (NN) has been used in many areas with great success. When a NN's structure (Model) is given, during the training steps, the parameters of the model are determined using an appropriate criterion and an optimization algorithm…

In this chapter we provide a theoretically founded investigation of state-of-the-art learning approaches for inverse problems from the point of view of spectral reconstruction operators. We give an extended definition of regularization…

This is a tutorial and survey paper on metric learning. Algorithms are divided into spectral, probabilistic, and deep metric learning. We first start with the definition of distance metric, Mahalanobis distance, and generalized Mahalanobis…

In this work, we introduce a machine/deep learning methodology to solve parametric integrals. Besides classical machine learning approaches, we consider a differential learning framework that incorporates derivative information during…

Convolution-type integral equations commonly occur in signal processing and image processing. Discretizing these equations yields large and ill-conditioned linear systems. While the classic multigrid method is effective for solving linear…

This paper explores the integration of Diophantine equations into neural network (NN) architectures to improve model interpretability, stability, and efficiency. By encoding and decoding neural network parameters as integer solutions to…

Hybrid optical neural networks (HONNs) offload some electronic computation to optical preprocessors to achieve low-power and fast training and inference phases in machine learning tasks. Our contribution to the development of HONNs is a…

This overview is devoted to splitting methods, a class of numerical integrators intended for differential equations that can be subdivided into different problems easier to solve than the original system. Closely connected with this class…