The study of parameter-dependent partial differential equations (parametric PDEs) with countably many parameters has been actively studied for the last few decades. In particular, it has been well known that a certain type of parametric holomorphy of the PDE solutions allows the application of deep neural networks without encountering the curse of dimensionality. This paper aims to propose a general framework for verifying the desired parametric holomorphy by utilizing the bounds on parametric derivatives. The framework is illustrated with examples of parametric elliptic eigenvalue problems (EVPs), encompassing both linear and semilinear cases. As the results, it will be shown that the ground eigenpairs have the desired holomorphy. Furthermore, under the same conditions, improved bounds for the mixed derivatives of the ground eigenpairs are derived. These bounds are well known to take a crucial role in the error analysis of quasi-Monte Carlo methods.

翻译：过去几十年间，具有可数多个参数的参数依赖偏微分方程（参数PDE）的研究一直备受关注。特别地，人们早已认识到，PDE解具有某种特定类型的参数全纯性，使得深度神经网络的应用能够避免维度灾难。本文旨在提出一个通用框架，通过利用参数导数的界来验证所需的参数全纯性。该框架以参数椭圆特征值问题（EVP）为例进行说明，涵盖线性和半线性情形。结果表明，基态特征对具有所需的参数全纯性。此外，在相同条件下，推导出了基态特征对混合导数的改进界。众所周知，这些界在拟蒙特卡罗方法的误差分析中起着关键作用。