Solving Inverse Parametrized Problems via Finite Elements and Extreme Learning Networks

2026-02-16

AI Summary

This work addresses the challenge of efficiently solving high-dimensional parameter-dependent partial differential equations arising in control, inverse problems, and uncertainty quantification. The authors propose a novel reduced-order modeling framework that combines finite element discretization in the physical domain with surrogate modeling in the parameter space: classical interpolation is employed for low-dimensional parameters, while extreme learning machines (ELMs) are introduced for the first time to handle high-dimensional settings. The method preserves Sobolev regularity and offers theoretically guaranteed error bounds, significantly enhancing computational efficiency. Moreover, it provides rigorous reconstruction error estimates for inverse problems. Demonstrated in quantitative photoacoustic tomography, the approach achieves high-fidelity recovery of both parameters and potential functions at a fraction of the computational cost of conventional methods, all while maintaining provable error guarantees.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Stochastic Optimization
• Intelligent Robots: State Estimation

Application Category

• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs

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.

Problem

• inverse problems
• parameter-dependent PDEs
• quantitative photoacoustic tomography
• high-dimensional parameter spaces
• reconstruction

Innovation

• reduced-order modeling
• extreme learning machine
• parameter-dependent PDEs
• inverse problems
• finite element method