A Bayesian Filtering Approach for Learning Lagrangian Dynamics from Noisy Measurements

2026-06-30

AI Summary

This work proposes a continuous-time stochastic state-space model grounded in Lagrangian mechanics for partially observed and noisy physical systems. The approach parameterizes kinetic and potential energy using neural networks, models unknown external forces as Gaussian white noise, and uniquely integrates Bayesian filtering with Lagrangian neural networks to jointly estimate system states and dynamical parameters. By combining Gaussian approximations with maximum likelihood learning, the method significantly outperforms conventional Lagrangian neural networks in experiments on the simple pendulum and Duffing oscillator, and achieves performance approaching that of Bayesian filters with known dynamics—even when the true model is unavailable.

Abstract

This paper proposes a Bayesian filtering-based approach for learning the dynamics of a physical system from partial, noisy measurements. We model the system dynamics using a Lagrangian mechanics formulation. As in Lagrangian neural networks (LNNs), we parameterize the kinetic and potential energies with neural networks. The unknown external forces in the Lagrangian formulation are modeled as white Gaussian noise. The corresponding Euler--Lagrange equations then yield a continuous-time stochastic state-space model (SSM) that describes the system dynamics. The neural network parameters and system states are then jointly learned via a maximum-likelihood method using Gaussian-approximation-based Bayesian filters. The effectiveness of the proposed method is demonstrated on pendulum and Duffing oscillator examples, and its performance is compared with conventional LNNs and with approximate Bayesian filters using known system models.

Problem

• Lagrangian dynamics
• noisy measurements
• Bayesian filtering
• system identification
• neural networks

Innovation

• Bayesian filtering
• Lagrangian dynamics
• neural networks
• stochastic state-space model
• maximum-likelihood learning