Metaplectic Neural Operators 的通用逼近定理获得证明
Universal Approximation Theorem for Metaplectic Neural Operators
理论向的读者可以看这篇：作者证明了 metaplectic 神经算子的通用逼近性，只用一层隐藏算子层加一个非局部分支就够，覆盖 Fourier 和分数阶 Fourier 乘子构造。做算子学习或 PDE 求解的科研人员会关心这个数学基础。
一篇 arXiv 论文证明了 metaplectic neural operators（辛傅里叶类神经算子）在非空有界 Lipschitz 区域上的通用逼近性质。
作者假设激活函数光滑、全局 Lipschitz 且非多项式，得到了从 C^s 到 C^r 以及从 H^s 到 H^r 的连续算子在相应范数下的紧一致逼近，其中 s、r 为任意非负整数，输入输出通道维度为有限维。Sobolev 设定下，紧输入集还需在 L^\infty 中一致有界。
\n\n结构性上，单个隐藏算子层加一个非局部分支即可胜任：非局部分支是带零延拓与限制的全空间 metaplectic 乘子，关联任一固定的自由辛矩阵。标量符号原子和有限个位置特征由变换和区域预先确定，与目标算子、紧集和精度无关。
证明方法结合了平均神经算子理论与精确的 chirp 共轭平均恒等式及导数估计，结果涵盖 Fourier 乘子与固定的非退化分数阶 Fourier 乘子构造。
Universal Approximation Theorem for Metaplectic Neural Operators
We prove universal approximation for metaplectic neural operators on nonempty bounded Lipschitz domains. The nonlocal branches are whole-space metaplectic multipliers with zero extension and restriction, associated with any fixed free symplectic matrix. Assuming a smooth, globally Lipschitz, nonpolynomial activation, we obtain compact-uniform approximation of continuous operators from $C^s$ to $C^r$ and from $H^s$ to $H^r$ in the respective output norms, for arbitrary nonnegative integers $s,r$ and finite input and output channel dimensions. In the Sobolev setting, compact input sets must additionally be uniformly bounded in $L^\infty$. One hidden operator layer with one nonlocal branch suffices. A single scalar symbol atom and finitely many positional features are prescribed by the transform and domain independently of the target operator, compact set and accuracy. The proof combines averaging neural operator theory with an exact chirp-conjugated averaging identity and derivative estimates. The result includes Fourier and fixed nondegenerate fractional Fourier multiplier constructions.
本条情报归属 AITOP「论文」核心追踪栏目。涉及核心实体与技术议题包括：神经算子、Fourier乘子、通用逼近、Sobolev空间，读者可通过下方标签进入对应专题页查阅历史脉络与关联对比。
信源核验说明：本文原始线索采编自「arXiv cs.LG」（@Ahmed Abdeljawad, Malek Yacine Rezoug），经 AITOP 多维评分管线结构化提炼。