Optimization of Sparse Sensor Layouts and Data-Driven Reconstruction Methods for Steady-State and Transient Thermal Field Inverse Problems

Abstract

This paper investigates the inverse reconstruction of temperature fields under both steady-state and transient heat conduction scenarios. The central contribution lies in the structured development and validation of the Gappy Clustering-based Proper Orthogonal Decomposition (Gappy C-POD) method—an approach that, despite its conceptual origin alongside the clustering-based dimensionality reduction method guided by POD structures (C-POD), had previously lacked an explicit algorithmic framework or experimental validation. To this end, the study constructs a comprehensive solution framework that integrates sparse sensor layout optimization with data-driven field reconstruction techniques. Numerical models incorporating multiple internal heat sources and heterogeneous boundary conditions are solved using the finite difference method. Multiple sensor layout strategies—including random selection, S-OPT, the Correlation Coefficient Filtering Method (CCFM), and uniform sampling—are evaluated in conjunction with database generation techniques such as Latin Hypercube sampling, Sobol sequences, and maximum–minimum distance sampling. The experimental results demonstrate that both Gappy POD and Gappy C-POD exhibit strong robustness in low-modal scenarios (1–5 modes), with Gappy C-POD—when combined with the CCFM and maximum distance sampling—achieving the best reconstruction stability. In contrast, while POD-MLP and POD-RBF perform well at higher modal numbers (>10), they show increased sensitivity to sensor configuration and sample size. This research not only introduces the first complete implementation of the Gappy C-POD methodology but also provides a systematic evaluation of reconstruction performance across diverse sensor placement strategies and reconstruction algorithms. The results offer novel methodological insights into the integration of data-driven modeling and sensor network design for solving inverse temperature field problems in complex thermal environments.