## 2 Materials and Methods
### PINNs for the Electro-Thermal Coupling Problem  
In this section, we introduce the principle of solving a two-dimensional electro-thermal coupling problem with PINNs. The electrokinetic field problem seeks to find the electric field or current density distribution in a conducting region. The problem can be described by the electric scalar potential in the form of the Laplace equation. Considering \(u(x,y)\) as the space distribution of the electric potential, \(\sigma\) denotes the electric conductivity, \(\partial u/\partial\mathbf{n}\) denotes the normal derivative of \(u\) along the outward direction of the Neumann boundary \(\Gamma_{N}\), \(B(x,y)\) is the given value of \(u\) on the Dirichlet boundary \(\Gamma_{D}\), and \(u(x,y)\) satisfies the following partial differential equation and boundary conditions:  
\[\
abla\cdot\sigma\
abla u(x,y)=0, \tag{9}\]  
\[\frac{\partial u(x_{N},y_{N})}{\partial\mathbf{n}}|_{\Gamma_{N}}=0, \tag{10}\]  
\[u(x_{D},y_{D})|_{\Gamma_{D}}=B(x,y). \tag{11}\]  
Figure 2: A schematic plot of PINN. The electric current generates Joule heat, which causes the temperature change of the conductor. The calculation formula for Joule heat \(Q\) at each point is  
\[Q=\mathbf{J}\cdot\mathbf{E}. \tag{12}\]  
where \(\mathbf{J}\) denotes the electric current density and \(\mathbf{E}\) the electric field, which is expressed as \(\mathbf{E}=-\
abla u\). The constitutive equation is given by  
\[\mathbf{J}=\sigma\mathbf{E}. \tag{13}\]  
The thermal field induced by the Joule heat is in the form of steady-state heat conduction. Considering \(T\) as the spatial temperature distribution, \(k\) as the thermal conductivity, and \(Q\) as representing the Joule heat source, which can be computed by Equations (12) and (13), the governing equation and boundary condition is given by  
\[-\
abla\cdot k\
abla T(x,y)=Q, \tag{14}\]  
\[T(x_{D},y_{D})|_{\Gamma_{D}}=T_{0}. \tag{15}\]  
where \(T_{0}\) is set to \(273K\). To handle the electro-thermal coupling problem, we employ two PINN networks to approximate solutions separately and train the networks in a sequential manner. The flow chart is shown in Figure 3, both of the two neural networks' inputs are the same sampling points, and the outputs are the electric scalar potential and the temperature, denoted by \(u\) and \(T\), respectively. The corresponding loss functions of those two problems are listed below. Since we employ the \"hard\" boundary condition method here, the loss function for the Dirichlet boundary is eliminated. Equations (16) and (17) denote the PDE constraint and Neumann boundary constraint of the steady electric field problem, respectively. \(\hat{u}(x^{i}_{f},y^{i}_{f};\mathbf{\Theta})\) denotes the output of PINN corresponding to the domain sample points, and \(\hat{u}(x^{i}_{N},y^{i}_{N};\mathbf{\Theta})\) represents the output of the Neumann boundary sample points. \[MSE_{F}=\frac{1}{N_{f}}\sum_{i=1}^{N_{f}}|\frac{\partial}{\partial x}(\sigma \frac{\partial\hat{u}(x^{i}_{f},y^{i}_{f};\mathbf{\Theta})}{\partial x})+ \frac{\partial}{\partial y}(\sigma\frac{\partial\hat{u}(x^{i}_{f},y^{i}_{f}; \mathbf{\Theta})}{\partial y})|^{2}, \tag{16}\]  
\[MSE_{B}=\frac{1}{N_{B}}\sum_{i=1}^{N_{B}}|\frac{\partial\hat{u}(x^{i}_{N},y^{ i}_{N};\mathbf{\Theta})}{\partial\mathbf{n}}|^{2}. \tag{17}\]  
Equation (18) is the PDE constraint of the thermal field problem. \(\hat{T}(x^{i}_{f},y^{i}_{f};\mathbf{\Theta})\) is the output of PINN corresponding to the temperature domain sample points. No \(MSE_{B}\) term is needed since the thermal field problem only has Dirichlet boundary conditions. \[\frac{\partial u(x_{N},y_{N})}{\partial\mathbf{n}}|_{\Gamma_{N}}=0, \tag{10}\]0  
Firstly, we train the neural network of the steady electric field problem, where the loss function is embedded with the governing equation and boundary conditions of Equations (9)-(11). When the loss function drops below a designated cut-off threshold or when the iteration reaches a specified number, the PINN training ends.