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 a Regression-based Physics-Informed Neural Networks (Reg-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出令人瞩目的结果。在本研究中，我们提出了一种基于回归的物理信息神经网络（Reg-PINNs），它将基于物理的数值计算与普通机器学习相结合。这一新一代物理信息神经网络克服了以往方法局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促进神经网络收敛并增强其泛化能力。与Shue等人[1998]的模型相比，我们的模型均方根误差降低了约30%。本文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。