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：将神经网络融入常微分方程（ODEs）求解是计算数学中的一个关键研究方向。在神经网络架构中，融入常微分方程的内在结构具有增强预测能力与降低数据需求等优势。在这些结构化的常微分方程形式中，拉格朗日表示因其显著的物理基础而脱颖而出。基于这一框架，Bhattoo引入了拉格朗日神经网络（LNNs）的概念。本文在此基础上提出了一项突破性扩展——广义拉格朗日神经网络（Genralized Lagrangian Neural Networks），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重要性，我们基于广义拉格朗日方程构建了模型。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，我们开展了包括一维和二维示例在内的多项实验，并探讨了网络参数的影响，结果证明了广义拉格朗日神经网络（GLNNs）的优越性。