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[2602.14757] Solving Inverse Parametrized Problems via Finite Elements and Extreme Learning Networks
Summary
This paper presents a novel framework for solving inverse parametrized problems using finite element methods and extreme learning networks, focusing on computational efficiency and accuracy.
Why It Matters
The research addresses the challenges in solving parameter-dependent partial differential equations, which are crucial in fields like control theory and uncertainty quantification. By improving computational efficiency without sacrificing accuracy, this work has significant implications for applications in quantitative photoacoustic tomography and beyond.
Key Takeaways
- Introduces a reduced-order modeling framework for parameter-dependent PDEs.
- Establishes rigorous error estimates for spatial discretization and parameter approximation.
- Demonstrates the application of the framework in quantitative photoacoustic tomography.
- Shows that extreme learning machines can enhance computational efficiency in high-dimensional parameter spaces.
- Provides substantial computational savings compared to traditional methods.
Mathematics > Numerical Analysis arXiv:2602.14757 (math) [Submitted on 16 Feb 2026] Title:Solving Inverse Parametrized Problems via Finite Elements and Extreme Learning Networks Authors:Erik Burman, Mats G. Larson, Karl Larsson, Jonatan Vallin View a PDF of the paper titled Solving Inverse Parametrized Problems via Finite Elements and Extreme Learning Networks, by Erik Burman and 3 other authors View PDF HTML (experimental) 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 paramet...
