## 3 Experiments and Results  
### Electro-Thermal Coupling Problem  
In this section, the two deep neural networks whose loss functions are defined in Equation (4) are employed to study the electro-thermal coupling issue. The governing equations and boundary conditions of the electrokinetic problem are defined in Equations (9)-(11), and counterparts of the thermal problem are defined in Equations (14) and (15). The deep learning framework we choose is PyTorch. The configuration of the computer is Intel Core i7-9750H for CPU, Nvidia GTX1660 Ti 6G for GPU, with 16 GB RAM. We use PyTorch's automatic differentiation [36] to establish the governing equations and the Neumann boundary constraints. A Rectangle Electro-Thermal Coupling Problem  
We choose a model of a square geometry [\(-0.5\), \(0.5\)] \(\times\) [\(-0.5\), \(0.5\)] m\({}^{2}\). As a proof of concept, here, we take the electric conductivity and thermal conductivity as isotropic scalars, and we set \(\sigma=1\) S/m and \(k=1\) W/(m \(\cdot\) K). The sample results are shown in Figure 4. There are 100 sample points in the domain for both the electric and thermal field problems. Each field has different boundary conditions; for the steady electric field, the upside border of the model is applied to a voltage of 1 V, while the downside border is grounded and the Neumann boundary condition is applied to the lateral lines. Since we use the \"hard\" boundary method to process the Dirichlet boundary condition, the upper and lower boundaries do not need to be sampled for the electric field problem. As a result, only 40 points on the two lateral boundaries are sampled for the electric field problem. While for the thermal field, all the boundaries are Dirichlet boundaries, their boundary conditions are also automatically satisfied by the \"hard\" boundary method, and hence, there is no need to sample on these boundaries. The points in the inner region are sampled by the Latin hypercube sampling method [37], and the boundary data are sampled randomly in a uniform distribution. Figure 3: The flow diagram of electro-thermal coupling computation based on PINN. For our study, the electrokinetic field's governing equation is the Laplace equation. Here, we construct a \"hard\" boundary model adapted to the geometry and boundary conditions. The smooth extension function \(G(x,y)\) and the smooth distance function \(D(x,y)\) defined in Equation (8) are specifically defined as follows, respectively. \[G(x,y)=y+0.5, \tag{19}\]  
\[D(x,y)=(0.5^{2}-y^{2}). \tag{20}\]  
After the \"hard\" boundary condition is applied, since we treat the electric conductivity as isotropic scalar, the final PDE expressions of the electric field problem based on the raw NN output \(u(x,y;\mathbf{\Theta})\) are given by  
\[-\sigma(u\
abla^{2}D+2\
abla D\
abla u+D\
abla^{2}u+\
abla^{2}G)=0. \tag{21}\]  
Since the Dirichlet boundary is compulsorily satisfied, the loss term of the steady electric field network model only includes governing equations and constraints of the Neumann boundaries. When the training of the electric field is over, we can obtain a satisfactory electric field solution; then, the Joule heat is calculated according to the obtained result. Next, the heat source data are transferred to the thermal field to continue the calculation. For the heat conduction equation, we also construct \"hard\" boundary conditions. Since we set the temperature on the boundaries to \(273\)\(K\), we have \(G(x,y)=273\). The distance function is given by  
\[D(x,y)=(x^{2}-0.5^{2})(y^{2}-0.5^{2}). \tag{22}\]  
Similarly, the final PDE expression of the thermal field problem based on the raw NN output \(T(x,y;\mathbf{\Theta})\) is given by  
\[-k(T\
abla^{2}D+2\
abla D\
abla T+D\
abla^{2}T+\
abla^{2}G)=Q. \tag{23}\]  
Two separate PINNs are employed to approximate the steady electric field and the thermal field, respectively. After parameter tuning of the neural network, we can obtain a well-trained model. For both neural networks, we choose 6 hidden layers with 20 neurons in each layer.