# Learning mechanical systems from real-world data using discrete forced Lagrangian dynamics

arXiv: https://arxiv.org/abs/2505.20370 · HTML: https://mlsanity.com/paper/2505.20370/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 59 references that are in the map.
- First posted on arXiv 26 May 2025.

## Builds on (most cited first, 21 of 21)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Generalizing to Unseen Domains: A Survey on Domain Generalization](https://mlsanity.com/paper/2103.03097/) (2021)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [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)
- [Theory-guided Data Science: A New Paradigm for Scientific Discovery from Data](https://mlsanity.com/paper/1612.08544/) (2016)
- [Discovering physical concepts with neural networks](https://mlsanity.com/paper/1807.10300/) (2018)
- [Simplifying Hamiltonian and Lagrangian Neural Networks via Explicit Constraints](https://mlsanity.com/paper/2010.13581/) (2020)
- [Dissipative SymODEN: Encoding Hamiltonian Dynamics with Dissipation and Control into Deep Learning](https://mlsanity.com/paper/2002.08860/) (2020)
- [Port-Hamiltonian Neural Networks for Learning Explicit Time-Dependent Dynamical Systems](https://mlsanity.com/paper/2107.08024/) (2021)
- [Dissipative Hamiltonian Neural Networks: Learning Dissipative and Conservative Dynamics Separately](https://mlsanity.com/paper/2201.10085/) (2022)
- [Efficiently Parameterized Neural Metriplectic Systems](https://mlsanity.com/paper/2405.16305/) (2024)
- [Learning strange attractors with reservoir systems](https://mlsanity.com/paper/2108.05024/) (2021)
- [Learning Hamiltonians of constrained mechanical systems](https://mlsanity.com/paper/2201.13254/) (2022)
- [Pseudo-Hamiltonian neural networks for learning partial differential equations](https://mlsanity.com/paper/2304.14374/) (2023)
- [Generalized Lagrangian Neural Networks](https://mlsanity.com/paper/2401.03728/) (2024)
- [A Structure-Preserving Kernel Method for Learning Hamiltonian Systems](https://mlsanity.com/paper/2403.10070/) (2024)
- [Machine learning and serving of discrete field theories -- when artificial intelligence meets the discrete universe](https://mlsanity.com/paper/1910.10147/) (2019)
- [Discrete Lagrangian Neural Networks with Automatic Symmetry Discovery](https://mlsanity.com/paper/2211.10830/) (2022)
- [Learning Dynamical Systems from Noisy Data with Inverse-Explicit Integrators](https://mlsanity.com/paper/2306.03548/) (2023)
- [Symplectic Neural Networks Based on Dynamical Systems](https://mlsanity.com/paper/2408.09821/) (2024)

## Built on by (most cited first, 7 of 7)

- [Hypothesis-driven construction of mesoscopic dynamics](https://mlsanity.com/paper/2605.16211/) (2026)
- [Learning Dynamics from Input-Output Data with Hamiltonian Gaussian Processes](https://mlsanity.com/paper/2511.05330/) (2025)
- [Structure-Preserving Gaussian Processes Via Discrete Euler-Lagrange Equations](https://mlsanity.com/paper/2605.06246/) (2026)
- [Learning Forced Multibody Dynamics on Lie Groups](https://mlsanity.com/paper/2607.12627/) (2026)
- [LaGSplat: Inferring Physics-Governed Interactive Simulation from Monocular Video Using Latent Lagrangian Gaussian Splatting](https://mlsanity.com/paper/2608.16324/) (2026)
- [Learning Deterministic and Stochastic Forced Hamiltonian Systems](https://mlsanity.com/paper/2608.19688/) (2026)
- [Dissipative Lagrangian Neural Networks for the Interpretable Modelling of Passive Mechanical Systems](https://mlsanity.com/paper/2610.15996/) (2026)

## Abstract (arXiv)

We introduce a data-driven method for learning the equations of motion of mechanical systems directly from position measurements, without requiring access to velocity data. This is particularly relevant in system identification tasks where only positional information is available, such as motion capture, pixel data or low-resolution tracking. Our approach takes advantage of the discrete Lagrange-d'Alembert principle and the forced discrete Euler-Lagrange equations to construct a physically grounded model of the system's dynamics. We decompose the dynamics into conservative and non-conservative components, which are learned separately using feed-forward neural networks. In the absence of external forces, our method reduces to a variational discretization of the action principle naturally preserving the symplectic structure of the underlying Hamiltonian system. We validate our approach on a variety of synthetic and real-world datasets, demonstrating its effectiveness compared to baseline methods. In particular, we apply our model to (1) measured human motion data and (2) latent embeddings obtained via an autoencoder trained on image sequences. We demonstrate that we can faithfully reconstruct and separate both the conservative and forced dynamics, yielding interpretable and physically consistent predictions.