A physics-driven sensor placement optimization methodology for temperature field reconstruction

Abstract

Perceiving the global field from sparse sensors has been a grand challenge in the monitoring, analysis, and design of physical systems. In this context, sensor placement optimization is a crucial issue. Most existing works require large and sufficient data to construct data-based criteria, which are intractable in data-free scenarios without numerical and experimental data. To this end, we propose a novel physics-driven sensor placement optimization (PSPO) method for temperature field reconstruction using a physics-based criterion to optimize sensor locations. In our methodological framework, we firstly derive the theoretical upper and lower bounds of the reconstruction error under noise scenarios by analyzing the optimal solution, proving that error bounds correlate with the condition number determined by sensor locations. Furthermore, the condition number, as the physics-based criterion, is used to optimize sensor locations by the genetic algorithm. Finally, the best sensors are validated by reconstruction models, including non-invasive end-to-end models, non-invasive reduced-order models, and physics-informed models. Experimental results, both on a numerical and an application case, demonstrate that the PSPO method significantly outperforms random and uniform selection methods, improving the reconstruction accuracy by nearly an order of magnitude. Moreover, the PSPO method can achieve comparable reconstruction accuracy to the existing data-driven placement optimization methods.

Figures (12)

• Figure 1: Conceptual flow of the PSPO method for temperature field reconstruction, where MLP and DeepOnet belong to non-invasive end-to-end models, PODNN belongs to the non-invasive reduced-order model, and PINN belongs to physics-informed model.
• Figure 2: 1D heat equation: comparisons of MLP, DeepOnet, PODNN, and PINN using PSPO, RS, and US under three noise scales. The first two rows represent Max-AEs and MSEs. The columns (from left to right) represent the results of MLP, DeepOnet, PODNN, and PINN, respectively. The black circle represents an outlier in a box plot. Error bar (square) represents results using 100 strategies of sensor placement randomly generated by RS.
• Figure 3: 1D heat equation: predictions of MLP, DeepOnet, PODNN, and PINN using PSPO, RS, and US under $\lambda_1=0.49$, $\lambda_2=2.25$, $\varepsilon_u \sim \mathcal{N}\left(0,0.1^2\right)$ and $k=10$.
• Figure 4: 1D heat equation: comparisons of MLP, DeepOnet, PODNN, and PINN with PSPO, US, CNS, and ECS under the different sensor number. The row represents MAx-AEs and MSEs. The columns (from left to right) represent results of MLP, DeepOnet, PODNN, and PINN, respectively.
• Figure 5: Application case: a signal processing module. Sensors are placed over the process plate to monitor the global temperature field.