Approximation of Maximally Monotone Operators : A Graph Convergence Perspective
Provides a new theoretical foundation for operator learning in settings where classical uniform or L^p approximation fails, relevant to PDEs and inverse problems.
The paper introduces graph convergence as a framework for approximating discontinuous and set-valued operators, proving that maximally monotone operators can be approximated by continuous encoder-decoder architectures while preserving structure.
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.