###### Abstract  
The problem of electro-thermal coupling is widely present in the integrated circuit (IC). The accuracy and efficiency of traditional solution methods, such as the finite element method (FEM), are tightly related to the quality and density of mesh construction. Recently, PINN (physics-informed neural network) was proposed as a method for solving differential equations. This method is mesh free and generalizes the process of solving PDEs regardless of the equations' structure. Therefore, an experiment is conducted to explore the feasibility of PINN in solving electro-thermal coupling problems, which include the electrokinetic field and steady-state thermal field. We utilize two neural networks in the form of sequential training to approximate the electric field and the thermal field, respectively. The experimental results show that PINN provides good accuracy in solving electro-thermal coupling problems. electro-thermal coupling; deep learning; physics-informed neural network; PDEs +
Footnote †: journal: _arxiv. used to determine the error between the output of neural network and a given target value, are embedded with some physical constraints. The deep learning training process could be directed to minimize the governing equations' residual. Physical constraints can compensate for the lack of data so that less training data are needed to approximate the solution of the equation. This idea of physically constrained deep learning dates back to the 1990s [7; 8; 9]. Owing to NVIDIA's breakthrough in GPU computing and the perfection of deep learning theory, this idea was recently revived. [PERSON] et al. [10] proposed a mesh-free deep Galerkin method (DGM) based on deep learning, and solved high-dimensional PDEs, including the Hamilton-Jacobi-Bellman PDE and the Burgers' equation. [PERSON] et al. [11] and [PERSON] et al. [12] leveraged deep neural networks to solve interface problems in domains with multiple materials. [PERSON] et al. [13] focused on the approximation of PDEs with complex geometries. [PERSON] et al. [14] utilized deep learning to predict low frequency seismic data. [PERSON] et al. [15] proposed a ML-decent method to learn the optimization algorithm and applied it to full-waveform inversion problem. Recently, [PERSON] et al. [16] formulated a deep learning method under physical constraints as the physics-informed neural networks (PINNs), which makes the neural network an interpretable optimization problem. With the concept of automatic differentiation, spatial and temporal coordinates' partial derivatives can be conveniently calculated using the weights and bias of the network, and then are included in the loss function [17]. Additionally, this method has the advantages of being mesh free and being usable for solving PDEs regardless of the equations' structure or even their complexity [18]. Moreover, the training points can be randomly sampled from the domain of interest, and less training data samples are required, compared with data-driven methods, avoiding the curse of dimensionality. The trained neural network is an analytical approximation of the latent solution, and hence, no interpolation is required for the solution at new input points [1], and further computation of derivatives, such as gradients, is convenient and smooth. [PERSON] employed PINN to solve one-dimensional Burger partial differential equations and the inverse problem of 2D/3D PDEs. After that, research works based on the PINN mushroomed. [PERSON] et al. [19; 20] proposed a parallel calculation method, which decomposes the domain into a few subdomains, then uses separate networks to represent each subdomains' function. [PERSON] et al. [21] leveraged PINN to solve the frequency-domain wave equation and introduced an adaptive sinusoidal activation function to improve the training performance. [PERSON] et al. [22] developed a Python toolkit based on Tensorflow, which is named DeepXDE and integrates the application of the PINN. [PERSON] et al. [23] studied the three-dimensional time-independent surface problem.