# A Bayesian Filtering Approach for Learning Lagrangian Dynamics from Noisy Measurements

arXiv: https://arxiv.org/abs/2606.31137 · HTML: https://mlsanity.com/paper/2606.31137/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 31 references that are in the map.
- First posted on arXiv 30 Jun 2026.

## Builds on (most cited first, 11 of 11)

- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [Lagrangian Neural Networks](https://mlsanity.com/paper/2003.04630/) (2020)
- [Deep Lagrangian Networks: Using Physics as Model Prior for Deep Learning](https://mlsanity.com/paper/1907.04490/) (2019)
- [Deep Lagrangian Networks for end-to-end learning of energy-based control for under-actuated systems](https://mlsanity.com/paper/1907.04489/) (2019)
- [Combining Physics and Deep Learning to learn Continuous-Time Dynamics Models](https://mlsanity.com/paper/2110.01894/) (2021)
- [Computationally Efficient Bayesian Learning of Gaussian Process State Space Models](https://mlsanity.com/paper/1506.02267/) (2015)
- [Sigma-Point Filtering and Smoothing Based Parameter Estimation in Nonlinear Dynamic Systems](https://mlsanity.com/paper/1504.06173/) (2015)
- [Nonlinear State Space Model Identification Using a Regularized Basis Function Expansion](https://mlsanity.com/paper/1510.00563/) (2015)
- [The Use of Gaussian Processes in System Identification](https://mlsanity.com/paper/1907.06066/) (2019)
- [Integrating Lagrangian Neural Networks into the Dyna Framework for Reinforcement Learning](https://mlsanity.com/paper/2603.08468/) (2026)

## Abstract (arXiv)

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.