Approximation of Maximally Monotone Operators : A Graph Convergence Perspective

TL;DR

This paper introduces a graph convergence framework for approximating maximally monotone operators, demonstrating that neural architectures can effectively approximate and preserve their structure.

Contribution

It proposes a novel graph convergence approach for operator approximation, showing neural networks can approximate maximally monotone operators while preserving their properties.

Findings

Neural networks can approximate maximally monotone operators via local graph convergence.

Uniform and L^p approximation are inadequate for these operators.

Structure-preserving approximations retain maximal monotonicity.

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 (Painlev\'e-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.