The ultimate goal of studying the magnetopause position is to accurately determine its location. Both traditional empirical computation methods and the currently popular machine learning approaches have shown promising results. In this study, we propose an Empirical Physics-Informed Neural Networks (Emp-PINNs) that combines physics-based numerical computation with vanilla machine learning. This new generation of Physics Informed Neural Networks overcomes the limitations of previous methods restricted to solving ordinary and partial differential equations by incorporating conventional empirical models to aid the convergence and enhance the generalization capability of the neural network. Compared to Shue et al. [1998], our model achieves a reduction of approximately 30% in root mean square error. The methodology presented in this study is not only applicable to space research but can also be referenced in studies across various fields, particularly those involving empirical models.

翻译：研究磁层顶位置的最终目标是精确确定其位置。传统的经验计算方法与当前流行的机器学习方法均展现了良好效果。本研究提出了一种结合基于物理的数值计算与原始机器学习的经验物理信息神经网络（Emp-PINNs）。这一新一代物理信息神经网络打破了以往方法仅限于求解常微分方程和偏微分方程的局限，通过引入传统经验模型来辅助神经网络收敛并增强其泛化能力。与Shue等人[1998]的方法相比，我们的模型均方根误差降低了约30%。本研究提出的方法不仅适用于空间研究，还可为涉及经验模型的各种领域研究提供参考。