# Lagrangian operator inference enhanced with structure-preserving machine learning for nonintrusive model reduction of mechanical systems

arXiv: https://arxiv.org/abs/2404.05040 · HTML: https://mlsanity.com/paper/2404.05040/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Apr 2024.

## Built on by (most cited first, 16 of 16)

- [A Riemannian Framework for Learning Reduced-order Lagrangian Dynamics](https://mlsanity.com/paper/2410.18868/) (2024)
- [Nested Operator Inference for Adaptive Data-Driven Learning of Reduced-order Models](https://mlsanity.com/paper/2508.11542/) (2025)
- [Physically consistent predictive reduced-order modeling by enhancing Operator Inference with state constraints](https://mlsanity.com/paper/2502.03672/) (2025)
- [Nonlinear energy-preserving model reduction with lifting transformations that quadratize the energy](https://mlsanity.com/paper/2503.02273/) (2025)
- [Learning Hamiltonian Dynamics at Scale: A Differential-Geometric Approach](https://mlsanity.com/paper/2509.24627/) (2025)
- [Streaming Operator Inference for Model Reduction of Large-Scale Dynamical Systems](https://mlsanity.com/paper/2601.12161/) (2026)
- [Data-driven Model Reduction for Soft Robots via Lagrangian Operator Inference](https://mlsanity.com/paper/2407.08840/) (2024)
- [NN-OpInf: an operator inference approach using structure-preserving composable neural networks](https://mlsanity.com/paper/2603.08488/) (2026)
- [Structure-preserving Lift & Learn: Scientific machine learning for nonlinear conservative partial differential equations](https://mlsanity.com/paper/2507.00301/) (2025)
- [Variational Neural Networks for Observable Thermodynamics (V-NOTS)](https://mlsanity.com/paper/2509.09899/) (2025)
- [Active learning for data-driven reduced models of parametric differential systems with Bayesian operator inference](https://mlsanity.com/paper/2601.00038/) (2025)
- [A neural operator for predicting vibration frequency response curves from limited data](https://mlsanity.com/paper/2603.10149/) (2026)
- [An ALE-Consistent Graph Neural Operator-Transformer Framework for Fluid-Structure Interaction](https://mlsanity.com/paper/2605.00937/) (2026)
- [Latent Lie-Poisson Neural Networks (LLPNNs): Discovering the motion of Lie-Poisson systems through observable data and latent dynamics](https://mlsanity.com/paper/2607.28939/) (2026)
- [LaGSplat: Inferring Physics-Governed Interactive Simulation from Monocular Video Using Latent Lagrangian Gaussian Splatting](https://mlsanity.com/paper/2608.16324/) (2026)
- [Latent-Lagrangian Neural Networks for Reduced Order Modeling of Non-autonomous Nonlinear Dynamical Systems](https://mlsanity.com/paper/2610.04723/) (2026)

## Abstract (arXiv)

Complex mechanical systems often exhibit strongly nonlinear behavior due to the presence of nonlinearities in the energy dissipation mechanisms, material constitutive relationships, or geometric/connectivity mechanics. Numerical modeling of these systems leads to nonlinear full-order models that possess an underlying Lagrangian structure. This work proposes a Lagrangian operator inference method enhanced with structure-preserving machine learning to learn nonlinear reduced-order models (ROMs) of nonlinear mechanical systems. This two-step approach first learns the best-fit linear Lagrangian ROM via Lagrangian operator inference and then presents a structure-preserving machine learning method to learn nonlinearities in the reduced space. The proposed approach can learn a structure-preserving nonlinear ROM purely from data, unlike the existing operator inference approaches that require knowledge about the mathematical form of nonlinear terms. From a machine learning perspective, it accelerates the training of the structure-preserving neural network by providing an informed prior, and it reduces the computational cost of the network training by operating on the reduced space. The method is first demonstrated on two simulated examples: a conservative nonlinear rod model and a two-dimensional nonlinear membrane with nonlinear internal damping. Finally, the method is demonstrated on an experimental dataset consisting of digital image correlation measurements taken from a lap-joint beam structure from which a predictive model is learned that captures amplitude-dependent frequency and damping characteristics accurately. The numerical results demonstrate that the proposed approach yields generalizable nonlinear ROMs that exhibit bounded energy error, capture the nonlinear characteristics reliably, and provide accurate long-time predictions outside the training data regime.