## 1 Introduction  
Engineering thermal management (ETM) is one of the critical tasks for quality control and system surveillance in many industries. As a prerequisite in ETM, the temperature field distribution should be known. Generally, estimation of a temperature field is of great interest, as accurately estimating a temperature field is required in a variety of research domains, such as ecology [1, 2], health care [3, 4], and meteorology [5, 6]. An accurate estimation of a temperature field may provide information for the guidance of the optimum design of engineering structures to reduce production cost and conserve energy [7]. Affected by a number of extrinsic factors (e.g., ambient temperature) and intrinsic factors (e.g., nature currents and moisture migration), a temperature field is often regarded as a complex system with stochastic physical dynamics and spatiotemporally varying uncertainties. For example, the temperature field in a granary (Figure 1), in which grain temperature at each location varies over time, is considered as a spatiotemporally varying thermodynamic system. However, due to the unknown inner mechanism and varying boundary conditions, accurately estimating a temperature field with spatiotemporal stochastic dynamics remains a challenging task. In conventional field estimation methods, a temperature field is estimated by physics-specific models. A heat transfer model for a cubic granary was adopted by [PERSON] and [PERSON] [8] and [PERSON] and [PERSON] [9]. [PERSON], [PERSON], and [PERSON] [10] and [PERSON] et al. [11] developed an unsteady heat transfer model to characterize a temperature field of a cylindrical granary. [PERSON] and [PERSON] [12] introduced a mathematical approach to model the airflow and the thermal state in a granary. For the stored grain ecosystem, a 3D transient heat, mass, and momentum transfer model was developed by [PERSON] et al. [13] and [PERSON] et al. [14] to predict temperatures and moisture contents of stored grains. [PERSON] et al. [15] and [PERSON] [16] extended the [PERSON]'s and [PERSON]'s work by considering a species transfer model, in which data of ambient temperature and humidity, wind speed, and solar radiation were used. Conventional methods for field estimation are mostly achieved by the simulation of physical models, in which sensing observations collected from distributed sensor networks are not considered. These methods may lead to a discrepancy between model predictions and observations because of stochastic dynamics and uncertainties in the temperature fields. For instance, excessive moisture of grain may lead to local overheating of grain, which cannot be easily captured by conventional field estimation methods. With the development of sensor technologies, wireless sensor networks have emerged as a key tool to collect sensing data and enable the sensing data to measure the fields [17; 18; 19]. Many researchers have employed statistical methods for the estimation of temperature fields using sensing observations, which aim to capture stochastic dynamics and uncertainties of temperature fields. For instance, [PERSON], [PERSON], and [PERSON] developed a health core temperature estimation approach using data collected from wearable sensors [20]. [PERSON] et al. analyzed temperature fields of the stored grain in different warehouses using grain temperature sensing data [21]. Uncertainty quantification to temperature sensing data is an essential part for temperature field estimation. Some research has focused on data uncertainty quantification problems [22; 23] For example, a data uncertainty quantification approach proposed by [PERSON] is for estimating temperature fields in the ocean [24]. [PERSON] and [PERSON] employed dynamically orthogonal field equations to characterize continuous dynamical systems [25]. Some other research work has focused on tackling uncertainties using Gaussian processes. For example, [PERSON] presented the application of Gaussian processes in machine learning [26].