Leray-Schauder Mappings for Operator Learning

TL;DR

This paper introduces a novel operator learning algorithm using Leray-Schauder mappings, capable of approximating nonlinear operators in Banach spaces, and demonstrates its effectiveness on benchmark datasets.

Contribution

It proposes a new operator learning method based on Leray-Schauder mappings, extending the class of universal approximators for nonlinear operators.

Findings

Achieves results comparable to state-of-the-art models on benchmarks.

Demonstrates efficiency in approximating nonlinear operators.

Provides a theoretical foundation for operator learning in Banach spaces.

Abstract

We present an algorithm for learning operators between Banach spaces, based on the use of Leray-Schauder mappings to learn a finite-dimensional approximation of compact subspaces. We show that the resulting method is a universal approximator of (possibly nonlinear) operators. We demonstrate the efficiency of the approach on two benchmark datasets showing it achieves results comparable to state of the art models.