Generalized Lagrangian Neural Networks

Incorporating neural networks for the inverse problem of 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 prediction accuracy 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 an extension (Generalized Lagrangian Neural Networks) to Lagrangian Neural Networks (LNNs) mainly based on mathematics and physics, 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 nonconservative 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).