Differentiable Sparse Identification of Lagrangian Dynamics

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.

Figures (4)

• Figure 1: Schematic architecture of differentiable sparse identification of lagrangian dynamics. (a) Measurement of the dynamical system and definition of the Lagrangian equations; (b) Description of equation formulations across different systems; (c) Loss functions under various system configurations; (d) Representation of values and their derivatives (first and second order) using cubic B-splines for each degree of freedom; (e) Schematic illustration of the sparse representation of the Euler-Lagrange equations; (f) Network is equivalent to solving optimization problem.
• Figure 2: Two types of dynamical systems are considered, as shown from left to right: active systems, including single pendulum, double pendulum, and spherical pendulum; and passive systems, including chaotic pendulum, cart-pendulum with a spring, spherical pendulum with a spring and magnetic pendulum.
• Figure 3: a Matches short-term; b Diverges long-term but preserves key statistical properties(amplitude range, oscillations). The basins of attraction for three magnets (colored red, blue, and green) are distributed along the edges of an equilateral triangle. c shows the true basin of attraction for magnetic pendulum, while d displays the basin obtained with the discovered parameters of system.
• Figure 4: Impact of Second-Order Derivative Regularization on Performance. a: Performance of our proposed model in fitting the measurements and their derivatives. b: Performance of the model without the regularization term.