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).

基于广义拉格朗日方程的变形不仅提高了预测准确性，而且在非保守系统中保证了拉格朗日表示，从而证明了广义拉格朗日神经网络（GLNNs）的优越性。