Projection Methods for Operator Learning and Universal Approximation

Jul 16, 2026

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A universal approximation theorem is established for continuous, possibly nonlinear, operators on arbitrary Banach spaces using the Leray-Schauder mapping. The work also develops an operator-learning method in Lp function spaces of multiple variables via orthogonal projections onto polynomial bases. Additional assumptions yield universal approximation results by learning a linear projection combined with a finite-dimensional mapping. For p=2, sufficient conditions ensure the approximation theorems. The framework supports deep learning approaches to operator learning.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages17

Reading Time43 min

Introduction

Nonlinear Projections for Operator Learning in Banach spaces

Learning Linear Projections on Banach Spaces of functions

Learning Linear Projections on the Hilbert space

Approximations for Fixed Points

Conclusions and Future Perspectives

arXiv :2406 . 12264v5 [math .NA] 9 Jul 2026 PROJECTION METHODS FOR OPERATOR LEARNING AND UNIVERSAL APPROXIMATION EMANUELE ZAPPALA Abstract. We obtain a new universal approximation theorem for continuous (possibly nonlinear) operators on arbitrary Banach spaces using the Leray-Schauder mapping. Moreover, we introduce and study a method for operator learning in Banach spaces Lp of functions with multiple variables, based on orthogonal projections on polynomial bases. We derive a universal approximation result for operators where we learn a linear projection and a finite dimensional mapping under some additional assumptions. For the case of p = 2, we give some sufficient conditions for the approximation results to hold. This article serves as the theoretical framework for a deep learning methodology in operator learning. MSC: 68T07 (primary); 46B28 (secondary) Keywords: neural operator; nonlinear operator; Leray-Schauder mapping. Contents 1. Introduction 1 2. Nonlinear Projections for Operator Learning in Banach spaces 5 3. Learning Linear Projections on Banach Spaces of functions 8 4. Learning Linear Projections on the Hilbert space 11 5. Approximations for Fixed Points 12 6. Conclusions and Future Perspectives 14 References 15 1. Introduction Operator learning is a branch of deep learning involved with approximating (potentially highly nonlinear) continuous operators between Banach spaces. The interest of operator learning lies in the fact that it allows to model complex phenomena, e.g. dynamical systems, whose underlying governing equations are not known [11,22,32,33] . The study of operator learning was initiated by the theoretical work [4], whose implementation was given in [22] . Since then, this field has expanded significantly both in its theoretical and applied scope to encompass a variety of architectures [5,19–21,25,28,31–33] Projection methods, e.g. Galerkin methods, are approaches for finding solutions of an operator equation by approximating this on prescribed subspaces through a projection [6,15] . After projecting the operator equation on a subspace, it is not necessarily true that this equation has a solution. When projected solutions exist, upon increasing the dimension of the subspaces one would want the solutions to converge to a solution of the original 2 EMANUELE ZAPPALA | Model | Spaces | Hypotheses | Approximation | Known Proj | Known Bases | | --- | --- | --- | --- | --- | --- | | Leray-Schauder | Banach | Cont operator | Univ | ✓/× | ✓/× | | Lp Proj | Lp | Cont operator | Univ | ✓ | ✓ | | DeepONet [22] | Banach to uniform | Cont operator | Univ | × | × | | NIE [32] | Compact open top | Cont integral operator | Univ | × | × | | ANIE [32] | Compact open top | Cont integral operator | Univ | × | × | | Spectral NIE [34] | H¨older | Frech´et integral operator | Univ | ✓ | ✓ | | Spectral NO[5] | ? | ? | ? | ✓ | ✓ | | PO-CKAN [28] | ? | ? | ? | × | × | | OPNO [21] | H¨older to L2 | Cont operator | Univ | × | ✓ | | FEPINN [31] | H¨older to L2 | PD operator | ? | × | ✓ | | RFM [25] | H¨older to L2 | PD operator | ? | × | ✓ | | PCA-Net [2,8] | Hilbert space | µ-meas+ | Prob > 1 − δ | ✓ | × | | CNO [27] | H¨older to Lp | PD operator+ | Univ | × | × | | MGNO [20] | L2 | ? | ? | × | ✓ | | AMG [19] | L2 | linear integral operator | ? | × | ✓ | Table 1 . Summary of varoius neural operator architectures with their theoretical hypotheses of applicability, approximation capabilities, and construction of approximations for projection on finite dimensional subspaces equation. This is not necessarily the case. The main question of projection methods is whether projected solutions exist, and converge to a solution of the original non-projected equation. We can formulate the problem of operator learning in relation to projection methods as follows. We want to learn projections on (finite dimensional) subspaces, and a map between subspaces such that we can approximate a target operator

FAQs

What universal approximation result does the document derive?

It derives a universal approximation theorem for continuous, possibly nonlinear, operators between Banach spaces using the Leray-Schauder mapping.

How is operator learning performed on Lp spaces in this work?

It introduces operator learning in Lp of multivariable functions using orthogonal projections onto polynomial bases.

What special analysis is provided for the case p=2?

For p=2, the document provides sufficient conditions ensuring the approximation results hold.

Jul 16, 2026

Upload by Jiven