Related papers: Operator Inference for Elliptic Eigenvalue Problem…
Neural operators have been applied in various scientific fields, such as solving parametric partial differential equations, dynamical systems with control, and inverse problems. However, challenges arise when dealing with input functions…
This paper introduces an efficient tensor-vector product technique for the rapid and accurate approximation of integral operators within physics-informed deep learning frameworks. Our approach leverages neural network architectures to…
We consider the minimization or maximization of the $J$th largest eigenvalue of an analytic and Hermitian matrix-valued function, and build on Mengi et al. (2014, SIAM J. Matrix Anal. Appl., 35, 699-724). This work addresses the setting…
We introduce an efficient method for computing the Stekloff eigenvalues associated with the Helmholtz equation. In general, this eigenvalue problem requires solving the Helmholtz equation with Dirichlet and/or Neumann boundary condition…
Solving Partial Differential Equation (PDE) interface problems on varying domains is a critical task in design and optimization, yet it remains computationally prohibitive for traditional solvers. Although operator learning has shown…
Neural operators effectively solve PDE problems from data without knowing the explicit equations, which learn the map from the input sequences of observed samples to the predicted values. Most existing works build the model in the original…
In this paper we discuss spectral properties of operators associated with the least-squares finite element approximation of elliptic partial differential equations. The convergence of the discrete eigenvalues and eigenfunctions towards the…
Recent advances in representation learning reveal that widely used objectives, such as contrastive and non-contrastive, implicitly perform spectral decomposition of a contextual kernel, induced by the relationship between inputs and their…
Many classical Computer Vision problems, such as essential matrix computation and pose estimation from 3D to 2D correspondences, can be solved by finding the eigenvector corresponding to the smallest, or zero, eigenvalue of a matrix…
In this paper, we obtain eigenvalue estimates for a larger class of elliptic differential operators in divergence form on a bounded domain in a complete Riemannian manifold isometrically immersed in Euclidean space. As an application, we…
We introduce a new approach for solving forward systems of differential equations using a combination of splitting methods and physics-informed neural networks (PINNs). The proposed method, splitting PINN, effectively addresses the…
A lower semi-definite self-adjoint linear operator in a Hilbert space is taken whose discrete spectrum is not empty and comprises at least several eigenvalues $\lambda_{min}=\lambda_1\leqslant\ldots\leqslant\lambda_m<\sigma_{ess}$. The…
We propose an extended Fourier neural operator (FNO) architecture for learning state and linear quadratic additive optimal control of systems governed by partial differential equations. Using the Ehrenpreis-Palamodov fundamental principle,…
The Koopman operator provides a linear framework to study nonlinear dynamical systems. Its spectra offer valuable insights into system dynamics, but the operator can exhibit both discrete and continuous spectra, complicating direct…
Implicit neural representations (INRs) have emerged as a powerful tool for solving inverse problems in computer vision and computational imaging. INRs represent images as continuous domain functions realized by a neural network taking…
The classical development of neural networks has been primarily for mappings between a finite-dimensional Euclidean space and a set of classes, or between two finite-dimensional Euclidean spaces. The purpose of this work is to generalize…
The paper presents a priori error analysis of the shallow neural network approximation to the solution to the indefinite elliptic equation and and cutting-edge implementation of the Orthogonal Greedy Algorithm (OGA) tailored to overcome the…
We present a neural operator framework for solving inverse scattering problems. A neural operator produces a preliminary indicator function for the scatterer, which, after appropriate rescaling, is used as a regularization parameter within…
We consider a new algorithm in light of the min-max Collatz-Wielandt formalism to compute the principal eigenvalue and the eigenvector (eigen-function) for a class of positive Perron-Frobenius-like operators. Such operators are natural…
The global behavior of dynamical systems can be studied by analyzing the eigenvalues and corresponding eigenfunctions of linear operators associated with the system. Two important operators which are frequently used to gain insight into the…