用有限元与极限学习机联合求解参数化微分方程，提升逆问题计算效率。
Solving Inverse Parametrized Problems via Finite Elements and Extreme Learning Networks
- 物理域用有限元离散，参数域用插值或极限学习机逼近
- 高维参数下误差可控，计算速度比传统方法快得多
- 适用于光声层析成像等逆问题，适合需快速求解的工程场景
我们提出一种基于插值的建模框架，用于求解控制、反问题和不确定性量化中出现的参数依赖型偏微分方程。解在物理域通过有限元方法离散，而对有限维参数的依赖则单独近似。我们建立了参数解的存在性、唯一性和正则性，并推导出严格误差估计，明确量化了空间离散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率。在高维参数空间中，我们以极限学习机（ELM）代理模型替代经典插值，在显式近似与稳定性假设下获得误差界。该框架应用于定量光声层析成像的反问题，推导出势场与参数重构误差估计，并展示相比标准方法显著的计算节省，且不损失精度。
原文摘要 · Abstract (English)
We develop an interpolation-based 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.
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