Sensors play an important role in monitoring manufacturing processes and update their digital twins. However, the data transmission bandwidth and sensor placement limitations in the physical systems may not allow us to collect the amount or the type of data that we wish. Recently, a physics-based compressive sensing (PBCS) approach was proposed to monitor manufacturing processes and obtain high-fidelity information with the reduced number of sensors by incorporating physical models of processes in compressed sensing. It can recover and reconstruct complete three-dimensional temperature distributions based on some limited measurements. In this paper, a constrained orthogonal matching pursuit algorithm is developed for PBCS, where coherence exists between the measurement matrix and the basis matrix. The efficiency of recovery is improved by introducing a boundary-domain reduction approach, which reduces the size of PBCS model matrices during the inverse operations. The improved PBCS method is demonstrated with the measurement of temperature distributions in the cooling and real-time printing processes of fused filament fabrication.