# Approximation of Maximally Monotone Operators : A Graph Convergence Perspective

arXiv: https://arxiv.org/abs/2605.12301 · HTML: https://mlsanity.com/paper/2605.12301/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 23 of its 54 references that are in the map.
- First posted on arXiv 12 May 2026.

## Builds on (most cited first, 23 of 23)

- [Spectral Normalization for Generative Adversarial Networks](https://mlsanity.com/paper/1802.05957/) (2018)
- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [PDE-Net: Learning PDEs from Data](https://mlsanity.com/paper/1710.09668/) (2017)
- [Model Reduction and Neural Networks for Parametric PDEs](https://mlsanity.com/paper/2005.03180/) (2020)
- [Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1801.06637/) (2018)
- [Sorting out Lipschitz function approximation](https://mlsanity.com/paper/1811.05381/) (2018)
- [Physics-informed learning of governing equations from scarce data](https://mlsanity.com/paper/2005.03448/) (2020)
- [Approximating Continuous Functions by ReLU Nets of Minimal Width](https://mlsanity.com/paper/1710.11278/) (2017)
- [Operator Learning: Algorithms and Analysis](https://mlsanity.com/paper/2402.15715/) (2024)
- [A Mathematical Guide to Operator Learning](https://mlsanity.com/paper/2312.14688/) (2023)
- [DeepMoD: Deep learning for Model Discovery in noisy data](https://mlsanity.com/paper/1904.09406/) (2019)
- [Learning Maximally Monotone Operators for Image Recovery](https://mlsanity.com/paper/2012.13247/) (2020)
- [Nonlocality and Nonlinearity Implies Universality in Operator Learning](https://mlsanity.com/paper/2304.13221/) (2023)
- [Two-layer neural networks with values in a Banach space](https://mlsanity.com/paper/2105.02095/) (2021)
- [Approximation of Lipschitz Functions using Deep Spline Neural Networks](https://mlsanity.com/paper/2204.06233/) (2022)
- [Operator learning meets inverse problems: A probabilistic perspective](https://mlsanity.com/paper/2508.20207/) (2025)
- [CFO: Learning Continuous-Time PDE Dynamics via Flow-Matched Neural Operators](https://mlsanity.com/paper/2512.05297/) (2025)
- [Approximation theory for 1-Lipschitz ResNets](https://mlsanity.com/paper/2505.12003/) (2025)
- [New universal operator approximation theorem for encoder-decoder architectures](https://mlsanity.com/paper/2503.24092/) (2025)
- [Learning truly monotone operators with applications to nonlinear inverse problems](https://mlsanity.com/paper/2404.00390/) (2024)
- [Gradient Flows and Nonlinear Power Methods for the Computation of Nonlinear Eigenfunctions](https://mlsanity.com/paper/2105.08405/) (2021)
- [Eigenvalue Problems in L^∞: Optimality Conditions, Duality, and Relations with Optimal Transport](https://mlsanity.com/paper/2107.12117/) (2021)
- [Introduction to Nonlinear Spectral Analysis](https://mlsanity.com/paper/2506.08754/) (2025)

## Abstract (arXiv)

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 Lp 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.