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.

翻译：本文分析了无限维函数空间之间映射的深度替代模型的逼近速率，这些映射通常来自线性和非线性偏微分方程的数据到解映射。具体而言，我们研究了深度神经算子与广义多项式混沌算子作为无限维可分希尔伯特空间之间非线性全纯映射替代模型的逼近速率。假设算子输入和输出函数空间由稳定的仿射表示系统参数化。可容许的表示系统包括所考虑空间的标准正交基、里斯基或合适的紧框架。针对在包含待表达映射定义域和值域的可分希尔伯特空间尺度上作用的深度神经和谱算子替代模型，建立了具有有限索伯列夫或贝索夫正则性的代数表达速率界。我们通过环面上线性椭圆型偏微分方程的系数到解映射的表达速率界阐述了这些抽象概念。