# A Riemannian Framework for Learning Reduced-order Lagrangian Dynamics

arXiv: https://arxiv.org/abs/2410.18868 · HTML: https://mlsanity.com/paper/2410.18868/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 41 references that are in the map.
- First posted on arXiv 24 Oct 2024.

## Builds on (most cited first, 25 of 25)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Hyperbolic Neural Networks](https://mlsanity.com/paper/1805.09112/) (2018)
- [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)
- [Riemannian Adaptive Optimization Methods](https://mlsanity.com/paper/1810.00760/) (2018)
- [A Riemannian Network for SPD Matrix Learning](https://mlsanity.com/paper/1608.04233/) (2016)
- [Geoopt: Riemannian Optimization in PyTorch](https://mlsanity.com/paper/2005.02819/) (2020)
- [Simplifying Hamiltonian and Lagrangian Neural Networks via Explicit Constraints](https://mlsanity.com/paper/2010.13581/) (2020)
- [Unsupervised Learning of Lagrangian Dynamics from Images for Prediction and Control](https://mlsanity.com/paper/2007.01926/) (2020)
- [Hamiltonian-based Neural ODE Networks on the SE(3) Manifold For Dynamics Learning and Control](https://mlsanity.com/paper/2106.12782/) (2021)
- [Extending Lagrangian and Hamiltonian Neural Networks with Differentiable Contact Models](https://mlsanity.com/paper/2102.06794/) (2021)
- [Quadratic Approximation Manifold for Mitigating the Kolmogorov Barrier in Nonlinear Projection-Based Model Order Reduction](https://mlsanity.com/paper/2204.02462/) (2022)
- [Which priors matter? Benchmarking models for learning latent dynamics](https://mlsanity.com/paper/2111.05458/) (2021)
- [Vector-valued Distance and Gyrocalculus on the Space of Symmetric Positive Definite Matrices](https://mlsanity.com/paper/2110.13475/) (2021)
- [Combining Physics and Deep Learning to learn Continuous-Time Dynamics Models](https://mlsanity.com/paper/2110.01894/) (2021)
- [Learning Nonlinear Projections for Reduced-Order Modeling of Dynamical Systems using Constrained Autoencoders](https://mlsanity.com/paper/2307.15288/) (2023)
- [Lagrangian operator inference enhanced with structure-preserving machine learning for nonintrusive model reduction of mechanical systems](https://mlsanity.com/paper/2404.05040/) (2024)
- [Learning Hamiltonians of constrained mechanical systems](https://mlsanity.com/paper/2201.13254/) (2022)
- [Symplectic model reduction of Hamiltonian systems using data-driven quadratic manifolds](https://mlsanity.com/paper/2305.15490/) (2023)
- [Matrix Manifold Neural Networks++](https://mlsanity.com/paper/2405.19206/) (2024)
- [Preserving Lagrangian structure in nonlinear model reduction with application to structural dynamics](https://mlsanity.com/paper/1401.8044/) (2014)
- [Learning Contact Dynamics using Physically Structured Neural Networks](https://mlsanity.com/paper/2102.11206/) (2021)
- [Neural Autoencoder-Based Structure-Preserving Model Order Reduction and Control Design for High-Dimensional Physical Systems](https://mlsanity.com/paper/2312.06256/) (2023)
- [Physically Consistent Learning of Conservative Lagrangian Systems with Gaussian Processes](https://mlsanity.com/paper/2206.12272/) (2022)

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

- [Physics-Informed Latent Neural Operator for Real-time Predictions of time-dependent parametric PDEs](https://mlsanity.com/paper/2501.08428/) (2025)
- [GeoHNNs: Geometric Hamiltonian Neural Networks](https://mlsanity.com/paper/2507.15678/) (2025)
- [Learning Hamiltonian Dynamics at Scale: A Differential-Geometric Approach](https://mlsanity.com/paper/2509.24627/) (2025)
- [Nonlinear Model Order Reduction of Dynamical Systems in Process Engineering: Review and Comparison](https://mlsanity.com/paper/2506.12819/) (2025)
- [Context-dependent manifold learning: A neuromodulated constrained autoencoder approach](https://mlsanity.com/paper/2603.11673/) (2026)
- [LaGSplat: Inferring Physics-Governed Interactive Simulation from Monocular Video Using Latent Lagrangian Gaussian Splatting](https://mlsanity.com/paper/2608.16324/) (2026)
- [Partially Linear Autoencoders for Manifold Learning and Dimensionality Reduction](https://mlsanity.com/paper/2608.29867/) (2026)

## Abstract (arXiv)

By incorporating physical consistency as inductive bias, deep neural networks display increased generalization capabilities and data efficiency in learning nonlinear dynamic models. However, the complexity of these models generally increases with the system dimensionality, requiring larger datasets, more complex deep networks, and significant computational effort. We propose a novel geometric network architecture to learn physically-consistent reduced-order dynamic parameters that accurately describe the original high-dimensional system behavior. This is achieved by building on recent advances in model-order reduction and by adopting a Riemannian perspective to jointly learn a non-linear structure-preserving latent space and the associated low-dimensional dynamics. Our approach enables accurate long-term predictions of the high-dimensional dynamics of rigid and deformable systems with increased data efficiency by inferring interpretable and physically-plausible reduced Lagrangian models.