Contents I. Introduction
Realistic engineering problems are described by partial differential equations (PDEs) that lack analytical solutions and necessitate the use of numerical methods such as the finite element method. Recently, in the advent of machine learning, deep neural network (DNN) has been employed to solve scientific problems as an alternative to classical numerical methods. Acting as a universal approximator, supervised learning by DNN enables implicit representation of constitutive relationships between inputs and labeled outputs data [1], [2]. However, such an approach is mainly suitable for data-rich problems with limited or no mathematical models. Thus, in order to accommodate a priori physical descriptions of scientific problems while simultaneously maintaining the unique features of machine learning, the governing physics should be directly encoded into neural networks (NNs). Such a physics-informed neural network (PINN) approach was proposed in [3], [4], and [5] and has been intensively developed in recent years. The architecture of PINN is varied, including fully connected network (Fc-NN) [6], convolutional neural network (CNN) [7], [8], [9], [10], and recurrent neural network (RNN) [9], [11], [12]. The solutions of PDEs are approximated by enforcing the governing equations as the loss functions of the NN without adding the labeled data, namely, the unsupervised training. The parameters of NN are optimized during the training process by minimizing the loss. The success of PINNs relies on two key ingredients, namely, back-propagation and automatic differentiation (AD). While the former allows a robust and efficient means to optimize the network parameters, the latter determines the change of outputs with respect to the change of inputs, i.e., the partial derivatives in the physics-informed loss function. In addition, the use of conventional finite difference (FD) to approximate partial derivatives has been popular in recent years [13]. Although PINNs have been successfully applied in various applications, including heat transfer [14], [15], solid mechanics [16], and fluid dynamics [17], [18], [19], their use in electromagnetism is gaining attraction [20], [21]. PINN models with different configurations have been developed to solve Maxwell’s equations, exploiting their great ability to solve PDEs, as discussed in [6], [22], [23], and [24]. In [25], a hypernetwork incorporating PINN is proposed to efficiently solve parameterized 2-D magnetostatic in both direct and inverse settings. However, most approaches limit the analysis to linear cases, neglecting nonlinear (NL) material laws and hysteresis. The integration of magnetic hysteresis in PINN was first introduced in [20]. They employed the FD method to approximate temporal derivatives for both reversible and irreversible problems. This advancement significantly enhanced the approximation of magnetic fields considering hysteresis within the PINN framework. We aim at further investigating PINNs to solve EM problems with different material characteristics, while striving to achieve computational efficiency. In particular, we propose PINN to solve the 1-D NL magnetic diffusion equation for different configurations. With regard to previous works [20], where FD is used for temporal derivatives calculation, we use AD in case of saturation (reversible law) and an original combination of CNN with PINN and FDs in case of hysteresis. The structure of this article is organized as follows. In Section II, we introduce the 1-D magneto-quasi-static (MQS) problem (namely, the 1-D magnetic diffusion equation) with the considered material laws (linear, saturated, and hysteretic). The PINN implementation with its two variants is provided in Section III. In Section IV, the proposed approach is applied to different test cases and compared with classical FE results for the sake of validation. Conclusion is drawn in Section V.
