Differentiable Sparse Identification of Lagrangian Dynamics

2025-11-13

AI Summary

Identifying Lagrangian equations for complex mechanical systems—particularly those involving rational-function terms—remains challenging under measurement noise. Method: This paper proposes a robust equation discovery framework that integrates physical constraints with cubic B-spline approximation. It innovatively employs B-splines to differentiably and robustly approximate state variables and their higher-order derivatives, augmented by a recursive derivative computation scheme. Within a physics-informed sparse regression framework, the method explicitly embeds Lagrangian structural priors—even from limited data. Contribution/Results: The approach effectively suppresses noise-induced errors and circumvents the sensitivity of conventional sparse regression to rational nonlinearities. Evaluated on multiple nonlinear mechanical benchmarks, it achieves superior equation identification accuracy and robustness compared to state-of-the-art baselines, enabling accurate and interpretable physical law discovery even under high noise levels.

Technology Category

• Intelligent Robots: State Estimation
• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Responsible Web: Machine-in-the-loop, human agency and autonomy

Abstract

Data-driven discovery of governing equations from data remains a fundamental challenge in nonlinear dynamics. Although sparse regression techniques have advanced system identification, they struggle with rational functions and noise sensitivity in complex mechanical systems. The Lagrangian formalism offers a promising alternative, as it typically avoids rational expressions and provides a more concise representation of system dynamics. However, existing Lagrangian identification methods are significantly affected by measurement noise and limited data availability. This paper presents a novel differentiable sparse identification framework that addresses these limitations through three key contributions: (1) the first integration of cubic B-Spline approximation into Lagrangian system identification, enabling accurate representation of complex nonlinearities, (2) a robust equation discovery mechanism that effectively utilizes measurements while incorporating known physical constraints, (3) a recursive derivative computation scheme based on B-spline basis functions, effectively constraining higher-order derivatives and reducing noise sensitivity on second-order dynamical systems. The proposed method demonstrates superior performance and enables more accurate and reliable extraction of physical laws from noisy data, particularly in complex mechanical systems compared to baseline methods.

Problem

• Identifies governing equations from noisy data in nonlinear dynamics
• Addresses noise sensitivity in Lagrangian system identification methods
• Enables accurate physical law extraction from complex mechanical systems

Innovation

• Integrates cubic B-spline approximation for nonlinear dynamics
• Robust equation discovery with physical constraints integration
• Recursive derivative computation using B-spline basis functions