Related papers: Multiwavelet-based Operator Learning for Different…
Solving partial differential equations (PDEs) is a required step in the simulation of natural and engineering systems. The associated computational costs significantly increase when exploring various scenarios, such as changes in initial or…
We consider the algebra of mixed multidimensional integral operators. In particular, Fredholm integral operators of the first and second kind belongs to this algebra. For the piecewise constant kernels we provide an explicit representation…
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,…
Physics-informed neural networks (PINNs) have emerged as a transformative framework for addressing operator learning and inverse problems involving the Korteweg-de Vries (KdV) equation for internal solitary waves. By integrating physical…
We develop in this paper a multi-grade deep learning method for solving nonlinear partial differential equations (PDEs). Deep neural networks (DNNs) have received super performance in solving PDEs in addition to their outstanding success in…
Neural operator learning as a means of mapping between complex function spaces has garnered significant attention in the field of computational science and engineering (CS&E). In this paper, we apply Neural operator learning to the…
This work explores the application of deep operator learning principles to a problem in statistical physics. Specifically, we consider the linear kinetic equation, consisting of a differential advection operator and an integral collision…
Solving nonlinear partial differential equations (PDEs) using kernel methods offers a compelling alternative to traditional numerical solvers. However, the performance of these methods strongly depends on the choice of kernel. In this work,…
This paper studies the construction of a refinement kernel for a given operator-valued reproducing kernel such that the vector-valued reproducing kernel Hilbert space of the refinement kernel contains that of the given one as a subspace.…
Magnetic Resonance Imaging can produce detailed images of the anatomy and physiology of the human body that can assist doctors in diagnosing and treating pathologies such as tumours. However, MRI suffers from very long acquisition times…
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…
We prove a few representer theorems for a localised version of the regularised and multiview support vector machine learning problem introduced by H.Q. Minh, L. Bazzani, and V. Murino, Journal of Machine Learning Research, 17(2016) 1-72,…
Solving high-dimensional partial differential equations (PDEs) efficiently requires handling multi-scale features across varying resolutions. To address this challenge, we present the Multiwavelet-based Multigrid Neural Operator (M2NO), a…
Ordinary differential equation (ODE) is widely used in modeling biological and physical processes in science. In this article, we propose a new reproducing kernel-based approach for estimation and inference of ODE given noisy observations.…
In this paper, we investigate the applications of operator learning, specifically DeepONet, for solving nonlinear partial differential equations (PDEs). Unlike conventional function learning methods that require training separate neural…
This paper concerns the distributed training of nonlinear kernel machines on Map-Reduce. We show that a re-formulation of Nystr\"om approximation based solution which is solved using gradient based techniques is well suited for this,…
In recent years operator networks have emerged as promising deep learning tools for approximating the solution to partial differential equations (PDEs). These networks map input functions that describe material properties, forcing functions…
We provide a new analytical approach to operator splitting for equations of the type $u_t=Au+u u_x$ where $A$ is a linear differential operator such that the equation is well-posed. Particular examples include the viscous Burgers' equation,…
Designing universal artificial intelligence (AI) solver for partial differential equations (PDEs) is an open-ended problem and a significant challenge in science and engineering. Currently, data-driven solvers have achieved great success,…
Learning probabilistic surrogates for partial differential equations remains challenging in data-scarce regimes: neural operators require large amounts of high-fidelity data, while generative approaches typically sacrifice resolution…