Generalized Lagrangian Neural Networks
Shanshan Xiao, Jiawei Zhang, Yifa Tang
TL;DR
This work extends Lagrangian Neural Networks to non-conservative dynamics by formulating Generalized Lagrangian Neural Networks (GLNNs) based on generalized Lagrange's equations $ \frac{d}{dt}\left( \frac{\partial \mathscr{L}}{\partial \Dot{q}_{k}} \right) - \frac{\partial \mathscr{L}}{\partial q_{k}} = F_{k}$ with $\mathscr{L}=T-U$. GLNNs learn both the Lagrangian $\mathscr{L}$ and the non-conservative term $F$ as neural networks, and derive $\Ddot{q}$ via $ \Ddot{q} = (\nabla_{\Dot{q}} \nabla_{\Dot{q}}^{\top}\mathscr{L})^{-1}[\nabla_{q}\mathscr{L} - (\nabla_{q} \nabla_{\Dot{q}}^{\top}\mathscr{L})\Dot{q} + F]$, enabling phase-flow predictions. Through damped harmonic motion and a compound double pendulum with friction, GLNNs demonstrate improved energy accuracy and phase dynamics over baseline neural models, while hyper-parameter studies reveal practical guidelines on depth and width. The approach preserves the physical interpretation of the Lagrangian while accommodating dissipation, offering a principled, structure-preserving alternative for learning non-conservative dynamical systems. However, GLNNs incur higher training complexity and may exhibit energy rebound in some dissipative cases, indicating avenues for refinement and broader applicability.
Abstract
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).