We develop an interpolation-based reduced-order modeling framework for parameter-dependent partial differential equations arising in control, inverse problems, and uncertainty quantification. The solution is discretized in the physical domain using finite element methods, while the dependence on a finite-dimensional parameter is approximated separately. We establish existence, uniqueness, and regularity of the parametric solution and derive rigorous error estimates that explicitly quantify the interplay between spatial discretization and parameter approximation. In low-dimensional parameter spaces, classical interpolation schemes yield algebraic convergence rates based on Sobolev regularity in the parameter variable. In higher-dimensional parameter spaces, we replace classical interpolation by extreme learning machine (ELM) surrogates and obtain error bounds under explicit approximation and stability assumptions. The proposed framework is applied to inverse problems in quantitative photoacoustic tomography, where we derive potential and parameter reconstruction error estimates and demonstrate substantial computational savings compared to standard approaches, without sacrificing accuracy.

翻译：本文针对控制、反问题及不确定性量化领域中出现的参数依赖偏微分方程，提出了一种基于插值的降阶建模框架。该框架在物理域采用有限元方法对解进行离散化，同时对有限维参数的依赖性进行独立逼近。我们建立了参数化解的存在性、唯一性与正则性，并推导出严格误差估计，明确量化了空间离散化与参数逼近之间的相互作用。在低维参数空间中，经典插值方案基于参数变量的Sobolev正则性获得代数收敛速率。在高维参数空间中，我们采用极限学习机（ELM）代理模型替代经典插值方法，并在显式逼近与稳定性假设下获得误差界。所提框架应用于定量光声断层成像中的反问题，推导了势场与参数重建的误差估计，并在保持精度的前提下，相比标准方法实现了显著的计算效率提升。