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Leray-Schauder Mappings for Operator Learning

Emanuele Zappala

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.

Leray-Schauder Mappings for Operator Learning

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.
Paper Structure (4 sections, 5 theorems, 23 equations, 2 figures, 1 table, 1 algorithm)

This paper contains 4 sections, 5 theorems, 23 equations, 2 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Theoretical Preliminaries
  3. Algorithm
  4. Experiments

Key Result

Theorem 2.1

Let $X$ and $Y$ be Banach spaces, let $T: X\longrightarrow Y$ be a continuous (possibly nonlinear) map, and let $K\subset X$ be a compact subset. Then, for any choice of $\epsilon > 0$ there exist natural numbers $n,m\in \mathbb N$, finite dimensional subspaces $E_n \subset X$ and $E_m \subset Y$, c where $\phi_k : E_k \longrightarrow \mathbb R^k$ indicates an isomorphism between the finite dimens

Figures (2)

  • Figure 1: Example of ground truth data for Burgers' dynamics where $\vec{x}$-axis is time and $\vec{y}$-axis represents space
  • Figure 2: Example of model's prediction for Burgers' dynamics where $\vec{x}$-axis is time and $\vec{y}$-axis represents space

Theorems & Definitions (10)

  • Theorem 2.1
  • Theorem 2.2
  • proof
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • Theorem 2.5
  • proof
  • Remark 2.6