基于物理信息神经网络的电磁场计算方法

Electromagnetic Field Calculation Method Based on Physical Informed Neural Network

物理信息神经网络(PINNs)将偏微分方程(PDEs)及其定解条件编码进网络中,使PDEs残差最小化的同时逼近定解条件,实现PDEs的求解。由于电磁场计算时存在局部高梯度问题、含源方程引发的训练困难问题和高对比系数界面识别问题等,PINNs在用于电磁场方程求解时训练效率低、计算精度不高,因而目前应用较少。该文对于PINNs在电磁场中的训练困难问题进行理论分析,提出了针对电磁场PDEs形式和神经网络架构的修改方法,实现了基于PINNs的静电场和稳恒磁场求解,计算结果准确性较好。将该方法推广到方程更加复杂的频域涡流场求解中,求解结果表明PINNs可以在复杂的频域方程上保证良好的精度。该研究工作为实现电磁场快速计算提供了新思路。

In recent years,the advent of language models such as GPT and Sora has underscored the computational prowess of data-driven models within high-dimensional parameter spaces.This has positioned them as the forefront of electromagnetic optimization design,and serves the fast computation of electromagnetic fields.However,the efficacy of these data-driven models hinges significantly on labeled data,and they grapple with challenges such as overfitting and a lack of physical understanding.Diverging from natural language processing(NLP),physical fields are usually described by a set of partial differential equations,the emergence of physics informed neural networks(PINNs)addresses this gap.The core concept of PINNs involves incorporating control equations into the neural network's loss function.This integration ensures that the network output approximates boundary conditions while adhering to the control equations within the solution domain.Nevertheless,PINNs encounter hurdles such as local high gradient issues,training complexities arising from source equations,and difficulties in identifying interfaces with high-contrast coefficients in electromagnetic field computations.Consequently,the training efficiency and computational accuracy of PINNs in solving electromagnetic field equations remain suboptimal,limiting their current applications.In order to use PINNs for the stabilization training of electromagnetic fields,this paper firstly explores a method for handling zero-value boundary conditions in electromagnetic fields by employing a fully connected neural network architecture with hard boundaries.This approach effectively eliminates boundary loss in the total loss function.Taking electrostatic field equations with sources as an illustration,the L2 error using this method against the analytical solution is 9.15×10-6,with almost complete satisfaction of boundary conditions.Recognizing the neural network's inclination to prioritize low-frequency or large-scale features,the paper introduces embedded Fourier features to process network inputs.Additionally,an adaptive adjustment strategy for weight coefficients is proposed.To enhance convergence,an improved fully connected neural network framework is utilized,considering connections between the input layer and each hidden layer.This framework is exemplified with boundary conditions involving leapfrog,resulting in a substantially reduced loss of about 10-7 during training,with nearly zero internal error.To mitigate the gradient explosion issue in PINNs caused by high-order source terms during electromagnetic field training,this study formulates dimensionless equations for the electromagnetic field without prior conditions.These equations are incorporated into the neural network as a novel loss function.The relative L2 errorϵbetween the PINNs and the FEM solution is measured at 2.24×10-2 for the static magnetic problem.Expanding this training methodology to the frequency-domain eddy current field,the study tackles eddy current scenarios.Solving the corresponding differential-algebraic equations with PINNs reveals the distribution of vector magnetic potentials and electromagnetic loss density in the frequency-domain eddy current field.The proposed relative L2 errorεfor vector magnetic potentials between PINNs and FEM solution is 1.3×10-2.The resolved electromagnetic losses by PINNs closely match the FEM method,with an error of 3.3×10-2.In conclusion,this paper presents tailored solutions for typical challenges in applying electromagnetic fields with PINNs,significantly enhancing training efficiency,reducing costs,and introducing innovative approaches for swift electromagnetic field calculations.