## 2 Methods
### Physics-Specific Model for the Global Temperature Profile  
We employ a physics-specific model to describe the global temperature profile of a temperature field over time. To illustrate how to choose a physics-specific model for a specific temperature field, we take the grain temperature field as an example.  
We consider a temperature field in a cubic granary. To describe the global temperature profile in a cubic granary, location \(\mathbf{s}\) in Equation (1) is represented by \(\{x,\ y,\ z\}\) in Cartesian coordinates. We employ a 3D heat transmission in a granary as the mean function as follows:  
\[\frac{\partial g(\mathbf{s},\ t)}{\partial t}=\alpha_{x}\frac{\partial^{2}g( \mathbf{s},\ t)}{\partial x^{2}}+\alpha_{y}\frac{\partial^{2}g(\mathbf{s},\ t)}{ \partial y^{2}}+\alpha_{z}\frac{\partial^{2}g(\mathbf{s},\ t)}{\partial z^{2}}, \tag{2}\]  
where the parameters \(\alpha_{x}\), \(\alpha_{y}\), and \(\alpha_{z}\) are determined on the basis of thermal properties. For stored grains, \(\alpha_{x}\), \(\alpha_{y}\), and \(\alpha_{z}\) denote the thermal diffusivities of the grains in the \(x\), \(y\), and \(z\) directions, respectively. Alternatively, if the thermal diffusivities of the grains are unknown, we can calculate \(\alpha_{x}\), \(\alpha_{y}\), and \(\alpha_{z}\) using the proposed parameter estimation approach in [30].  
The finite-difference method is employed to solve Equation (2). We use an explicit marching method to calculate the global temperature profile \(g(\mathbf{s},t)\), as shown in Equation (3), where a central difference in \(x,y\), and \(z\) directions, and a forward difference in time \(t\) are considered. To be specific,  
\[\begin{array}{l}\frac{1}{\Delta t}\Big{(}g\Big{(}i_{x},i_{y},i_{z},i_{t}+1 \Big{)}-g\Big{(}i_{x},i_{y},i_{z},i_{t}\Big{)}\Big{)}\\ =\frac{\alpha_{x}}{\Delta x^{2}}\Big{(}g\Big{(}i_{x}+1,i_{y},i_{z},i_{t}\Big{)} -2g\Big{(}i_{x},i_{y},i_{z},i_{t}\Big{)}+g\Big{(}i_{x}-1,i_{y},i_{z},i_{t} \Big{)}\Big{)}\\ +\frac{\alpha_{y}}{\Delta y^{2}}\Big{(}g\Big{(}i_{x},i_{y}+1,i_{z},i_{t}\Big{)} -2g\Big{(}i_{x},i_{y},i_{z},i_{t}\Big{)}+g\Big{(}i_{x},i_{y}-1,i_{z},i_{t} \Big{)}\Big{)}\\ +\frac{\alpha_{y}}{\Delta z^{2}}\Big{(}g\Big{(}i_{x},i_{y},i_{z}+1,i_{t}\Big{)} -2g\Big{(}i_{x},i_{y},i_{z},i_{t}\Big{)}+g\Big{(}i_{x},i_{y},i_{z}-1,i_{t} \Big{)}\Big{)},\end{array} \tag{3}\]  
where \(i_{x}\), \(i_{y}\), \(i_{z}\), and \(i_{t}\) are the indexes of locations \(x\), \(y\), \(z\) and the time \(t\). \(\Delta x\), \(\Delta y\), \(\Delta z\), and \(\Delta t\) are the grid spacings in the \(x\), \(y\), and \(z\) directions of the space domain and time \(t\).  
To satisfy a highly accurate numerical solution, this method is stable under the condition [31]:  
\[\frac{\Delta t\alpha_{x}}{\Delta x^{2}}+\frac{\Delta t\alpha_{y}}{\Delta y^{2} }+\frac{\Delta t\alpha_{z}}{\Delta z^{2}}<0.5. \tag{4}\]  
Given the grain temperature at the initial time point and the temperature at the walls of the granary as the boundary conditions, the global temperature profiles are obtained. We adopt a Gaussian process model [26] for interpolation using the sensing data on the granary walls to obtain the boundary conditions.