Generalized Lagrangian Neural Networks

TL;DR

This paper introduces Generalized Lagrangian Neural Networks (GLNNs), extending Lagrangian Neural Networks to better model non-conservative systems, improving prediction accuracy and preserving physical structure.

Contribution

The paper presents a novel extension of Lagrangian Neural Networks tailored for non-conservative systems, based on the generalized Lagrange's equation, enhancing modeling capabilities.

Findings

GLNNs outperform previous models in accuracy.

Experiments on 1D and 2D systems validate effectiveness.

Network parameter analysis shows robustness.

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).