Spectral methods for Neural Integral Equations
Neural integral equations are deep learning models based on the theory of integral equations, where the model consists of an integral operator and the corresponding equation (of the second kind) which is learned through an optimization procedure. This approach allows to leverage the nonlocal properties of integral operators in machine learning, but it is computationally expensive. In this article, we introduce a framework for neural integral equations based on spectral methods that allows us to learn an operator in the spectral domain, resulting in a cheaper computational cost, as well as in high interpolation accuracy. We study the properties of our methods and show various theoretical guarantees regarding the approximation capabilities of the model, and convergence to solutions of the numerical methods. We provide numerical experiments to demonstrate the practical effectiveness of the resulting model.
Code (1)
Similar Papers 제목 키워드 기반
A level-wise training scheme for learning neural multigrid smoothers with application to integral equations
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 …
Ray-driven Spectral CT Reconstruction Based on Neural Base-Material Fields
In spectral CT reconstruction, the basis materials decomposition involves solving a large-scale nonlinear system of integral equations, which is highly ill-posed mathematically. This paper proposes a model that parameter…
CT ReconstructionNeural Integral Equations
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 equa…
Approximate numerical solutions of fractional integral equations using Laguerre and Touchard polynomials
Two numerical methods based on Laguerre and Touchard polynomials are de scribed in this paper to solve both the fractional integral equations of the first kind and the second kind (FIEs-1K and FIEs-2K, respectively). …
2kDeep Neural Network Solutions for Oscillatory Fredholm Integral Equations
We studied the use of deep neural networks (DNNs) in the numerical solution of the oscillatory Fredholm integral equation of the second kind. It is known that the solution of the equation exhibits certain oscillatory beh…