Related papers: Learning Partial Differential Equations in Reprodu…
A framework for coherent pattern extraction and prediction of observables of measure-preserving, ergodic dynamical systems with both atomic and continuous spectral components is developed. It is based on an approximation of the generator of…
A Hilbert space embedding of a distribution---in short, a kernel mean embedding---has recently emerged as a powerful tool for machine learning and inference. The basic idea behind this framework is to map distributions into a reproducing…
We study a functional linear regression model that deals with functional responses and allows for both functional covariates and high-dimensional vector covariates. The proposed model is flexible and nests several functional regression…
Spatiotemporal partial differential equations (PDEs) underpin a wide range of scientific and engineering applications. Neural PDE solvers offer a promising alternative to classical numerical methods. However, existing approaches typically…
We have studied possible applications of a particular pseudo-differential algebra in singular analysis for the construction of fundamental solutions and Green's functions of a certain class of elliptic partial differential operators. The…
We merge computational mechanics' definition of causal states (predictively-equivalent histories) with reproducing-kernel Hilbert space (RKHS) representation inference. The result is a widely-applicable method that infers causal structure…
We develop a mathematical framework to address a broad class of metric and preference learning problems within a Hilbert space. We obtain a novel representer theorem for the simultaneous task of metric and preference learning. Our key…
This is a tutorial and survey paper on kernels, kernel methods, and related fields. We start with reviewing the history of kernels in functional analysis and machine learning. Then, Mercer kernel, Hilbert and Banach spaces, Reproducing…
We study the problem of estimating linear response statistics under external perturbations using time series of unperturbed dynamics. Based on the fluctuation-dissipation theory, this problem is reformulated as an unsupervised learning task…
Deep Gaussian Processes (DGPs) leverage a compositional structure to model non-stationary processes. DGPs typically rely on local inducing point approximations across intermediate GP layers. Recent advances in DGP inference have shown that…
Partial Differential Equations (PDEs) describe phenomena ranging from turbulence and epidemics to quantum mechanics and financial markets. Despite recent advances in computational science, solving such PDEs for real-world applications…
We introduce a novel kernel learning framework toward efficiently solving nonlinear partial differential equations (PDEs). In contrast to the state-of-the-art kernel solver that embeds differential operators within kernels, posing…
We consider some non-linear non-homogeneous partial differential equations (PDEs) and derive their exact Green function solution as a functional Taylor expansion in powers of the source. The kind of PDEs we consider are dispersive ones…
In this paper we develop a way of obtaining Green's functions for partial differential equations with linear involutions by reducing the equation to a higher-order PDE without involutions. The developed theory is applied to a model of heat…
This paper reviews the functional aspects of statistical learning theory. The main point under consideration is the nature of the hypothesis set when no prior information is available but data. Within this framework we first discuss about…
In this paper we combine the theory of reproducing kernel Hilbert spaces with the field of collocation methods to solve boundary value problems with special emphasis on reproducing property of kernels. From the reproducing property of…
In this paper we consider the problems of supervised classification and regression in the case where attributes and labels are functions: a data is represented by a set of functions, and the label is also a function. We focus on the use of…
Learning convolution kernels in operators from data arises in numerous applications and represents an ill-posed inverse problem of broad interest. With scant prior information, kernel methods offer a natural nonparametric approach with…
Since its introduction, the Discrete Variable Representation (DVR) basis set has become an invaluable representation of state vectors and Hermitian operators in non-relativistic quantum dynamics and spectroscopy calculations. On the other…
In this article, the reproducing kernel Hilbert space [0, 1] is employed for solving a class of third-order periodic boundary value problem by using fitted reproducing kernel algorithm. The reproducing kernel function is built to get fast…