An Operator Learning Approach via Function-valued Reproducing Kernel Hilbert Space for Differential Equations

TL;DR

This paper introduces a neural network-based operator learning method utilizing function-valued reproducing kernel Hilbert spaces to efficiently solve both linear and nonlinear partial differential equations from limited data.

Contribution

It proposes a novel architecture that leverages function-valued RKHS for operator learning, enabling high-resolution solutions from low-resolution training data.

Findings

Achieves high accuracy on various PDEs with limited data

Capable of producing high-resolution solutions from low-resolution inputs

Demonstrates effectiveness on both linear and nonlinear PDEs

Abstract

Much recent work has addressed the solution of a family of partial differential equations by computing the inverse operator map between the input and solution space. Toward this end, we incorporate function-valued reproducing kernel Hilbert spaces in our operator learning model. We use neural networks to parameterize Hilbert-Schmidt integral operator and propose an architecture. Experiments including several typical datasets show that the proposed architecture has desirable accuracy on linear and nonlinear partial differential equations even with a small amount of data. By learning the mappings between function spaces, the proposed method can find the solution of a high-resolution input after learning from lower-resolution data.