Approximation rates are analyzed for deep surrogates of maps between infinite-dimensional function spaces, arising, e.g., as data-to-solution maps of linear and nonlinear partial differential equations. Specifically, we study approximation rates for deep neural operator and generalized polynomial chaos (gpc) Operator surrogates for nonlinear, holomorphic maps between infinite-dimensional, separable Hilbert spaces. Operator in- and outputs from function spaces are assumed to be parametrized by stable, affine representation systems. Admissible representation systems comprise orthonormal bases, Riesz bases, or suitable tight frames of the spaces under consideration. Algebraic expression rate bounds are established for both, deep neural and spectral operator surrogates acting in scales of separable Hilbert spaces containing domain and range of the map to be expressed, with finite Sobolev or Besov regularity. We illustrate the abstract concepts by expression rate bounds for the coefficient-to-solution map for a linear elliptic PDE on the torus.

分析无限维函数空间之间映射的深度代理的近似率，例如线性和非线性偏微分方程的数据到解映射。具体来说，我们研究了无限维、可分离希尔伯特空间之间的非线性、全纯映射的深度神经算子和广义多项式混沌 (gpc) 算子代理的近似率。假设函数空间的算子输入和输出由稳定的仿射表示系统参数化。可接受的表示系统包括正交基、Riesz 基或所考虑的空间的合适的紧框架。为在包含要表达的映射的域和范围的可分离希尔伯特空间尺度中起作用的深度神经和谱算子代理建立代数表达速率界限，具有有限的索博列夫或贝索夫正则性。我们通过环面上线性椭圆偏微分方程的系数解图的表达率界限来说明抽象概念。