We present a novel method for reconstructing the thermal conductivity coefficient in 1D and 2D heat equations using moving sensors that dynamically traverse the domain to record sparse and noisy temperature measurements. We significantly reduce the computational cost associated with forward PDE evaluations by employing automatic differentiation, enabling a more efficient and scalable reconstruction process. This allows the inverse problem to be solved with fewer sensors and observations. Specifically, we demonstrate the successful reconstruction of thermal conductivity on the 1D circle and 2D torus, using one and four moving sensors, respectively, with their positions recorded over time. Our method incorporates sampling algorithms to compute confidence intervals for the reconstructed conductivity, improving robustness against measurement noise. Extensive numerical simulations of heat dynamics validate the efficacy of our approach, confirming both the accuracy and stability of the reconstructed thermal conductivity. Additionally, the method is thoroughly tested using large datasets from machine learning, allowing us to evaluate its performance across various scenarios and ensure its reliability. This approach provides a cost-effective and flexible solution for conductivity reconstruction from sparse measurements, making it a robust tool for solving inverse problems in complex domains.

翻译：本文提出了一种新颖方法，用于重构一维和二维热传导方程中的热导率系数。该方法利用在域内动态移动的传感器记录稀疏且含噪声的温度测量数据。通过采用自动微分技术，我们显著降低了正问题偏微分方程求解的计算成本，实现了更高效、可扩展的重构过程，从而能够使用更少的传感器和观测数据求解该反问题。具体而言，我们成功演示了在一维圆环和二维环面上分别使用一个和四个移动传感器（其位置随时间被记录）进行热导率重构。本方法结合了采样算法以计算重构热导率的置信区间，从而增强了对测量噪声的鲁棒性。大量的热动力学数值模拟验证了本方法的有效性，证实了重构热导率的准确性和稳定性。此外，该方法利用来自机器学习的大规模数据集进行了全面测试，使我们能够评估其在各种场景下的性能并确保其可靠性。该方法为基于稀疏测量数据进行热导率重构提供了一种经济高效且灵活的解决方案，使其成为求解复杂域中反问题的强有力工具。