Solving Inverse Parametrized Problems via Finite Elements and Extreme Learning Networks
Erik Burman, Mats G. Larson, Karl Larsson, Jonatan Vallin
TL;DR
This work develops a reduction framework for parameterized PDEs that marries finite element discretization in the physical domain with a parameter-domain surrogate strategy. By proving existence, uniqueness, and regularity of the parametric solution and deriving joint error bounds, the approach separates spatial discretization from parameter approximation and adapts to both low- and high-dimensional parameter spaces. In the low-dimensional regime it leverages classical interpolation with algebraic convergence, while in the high-dimensional regime it adopts Extreme Learning Machines with Barron-space-based error controls, yielding dimension-independent rates under explicit assumptions. The framework is applied to inverse problems in quantitative photoacoustic tomography (QPAT), where potential and parameter reconstructions are bounded and validated with numerical experiments, demonstrating substantial computational savings over conventional methods without sacrificing accuracy. Its unified ROM structure supports fast optimization, uncertainty quantification, and multi-query tasks across diverse parameter regimes.
Abstract
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.