用图收敛方法逼近非连续算子，突破传统近似框架局限。
Approximation of Maximally Monotone Operators : A Graph Convergence Perspective
- 以图收敛代替传统逼近，适配闭算子的性质
- 连续编码器-解码器可局部逼近任意极大单调算子
- 通过基于预解算子的参数化，保持算子结构特性
算子学习在无限维空间间的连续映射（如偏微分方程求解算子）中已取得显著成功。然而，许多重要算子——包括微分算子——具有不连续或集值特性，超出了经典近似框架的适用范围。本文提出范式转变：采用图收敛（Painlevé-Kuratowski 收敛）来形式化算子逼近，该方法特别适用于闭算子。我们证明了统一逼近和 $L^p$ 逼近在此设定下本质上不足。聚焦于极大单调算子，我们证明任何此类算子均可通过连续编码器-解码器架构在局部图收敛意义下被逼近，并进一步构造出保留极大单调性的结构保持型近似，其核心为基于预解算子的参数化设计。
原文摘要 · Abstract (English)
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.
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