Approximation of Maximally Monotone Operators : A Graph Convergence Perspective

2026-05-12

AI Summary

This work addresses a fundamental limitation of conventional operator learning methods, which rely on uniform or $L^p$ approximation frameworks and struggle to handle discontinuous or set-valued operators—such as maximally monotone differential operators. To overcome this challenge, the paper introduces, for the first time, a novel approximation paradigm grounded in graph convergence (specifically, Painlevé–Kuratowski convergence). The proposed approach employs an encoder–decoder neural network architecture combined with resolvent-based parameterization to achieve continuous approximation in the sense of local graph convergence, while rigorously preserving maximal monotonicity. By doing so, this study transcends the constraints of classical approximation theory and establishes a structure-preserving theoretical framework for learning operators with inherent set-valued or discontinuous characteristics.

Abstract

Operator learning has been highly successful for continuous mappings between infinite-dimensional spaces, such as PDE solution operators. However, many operators of interest-including differential operators-are discontinuous or set-valued, and lie outside classical approximation frameworks. We propose a paradigm shift by formulating approximation via graph convergence (Painlevé-Kuratowski convergence), which is well-suited for closed operators. We show that uniform and $L^p$ approximation are fundamentally inadequate in this setting. Focusing on maximally monotone operators, we prove that any such operator can be approximated in the sense of local graph convergence by continuous encoder-decoder architectures, and further construct structure-preserving approximations that retain maximal monotonicity via resolvent-based parameterizations.

Problem

• maximally monotone operators
• operator approximation
• graph convergence
• discontinuous operators
• set-valued operators

Innovation

• graph convergence
• maximally monotone operators
• operator learning
• resolvent-based parameterization
• encoder-decoder architectures