Incorporating neural networks for the solution of Ordinary Differential Equations (ODEs) represents a pivotal research direction within computational mathematics. Within neural network architectures, the integration of the intrinsic structure of ODEs offers advantages such as enhanced predictive capabilities and reduced data utilization. Among these structural ODE forms, the Lagrangian representation stands out due to its significant physical underpinnings. Building upon this framework, Bhattoo introduced the concept of Lagrangian Neural Networks (LNNs). Then in this article, we introduce a groundbreaking extension (Genralized Lagrangian Neural Networks) to Lagrangian Neural Networks (LNNs), innovatively tailoring them for non-conservative systems. By leveraging the foundational importance of the Lagrangian within Lagrange's equations, we formulate the model based on the generalized Lagrange's equation. This modification not only enhances prediction accuracy but also guarantees Lagrangian representation in non-conservative systems. Furthermore, we perform various experiments, encompassing 1-dimensional and 2-dimensional examples, along with an examination of the impact of network parameters, which proved the superiority of Generalized Lagrangian Neural Networks(GLNNs).

翻译：将神经网络应用于常微分方程（ODEs）的求解是计算数学领域的关键研究方向。在神经网络架构中，融合ODEs的固有结构能带来预测能力增强、数据需求减少等优势。在这些结构化的ODE形式中，拉格朗日表示因其显著的物理基础而尤为突出。基于此框架，Bhattoo提出了拉格朗日神经网络（LNNs）。本文则进一步提出了一项开创性扩展——广义拉格朗日神经网络（GLNNs），创新性地将其应用于非保守系统。通过利用拉格朗日在拉格朗日方程中的基础性地位，我们基于广义拉格朗日方程构建模型。这一改进不仅提升了预测精度，还确保了非保守系统中拉格朗日表示的有效性。此外，我们开展了涵盖一维与二维示例的多种实验，并分析了网络参数的影响，实验结果证明了广义拉格朗日神经网络（GLNNs）的优越性。