# Data-driven Nonlinear Model Reduction to Spectral Submanifolds in Mechanical Systems

arXiv: https://arxiv.org/abs/2110.01929 · HTML: https://mlsanity.com/paper/2110.01929/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 91 references that are in the map.
- First posted on arXiv 5 Oct 2021.

## Builds on (most cited first, 7 of 7)

- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Data-driven discovery of Koopman eigenfunctions for control](https://mlsanity.com/paper/1707.01146/) (2017)
- [Data-Driven POD-Galerkin Reduced Order Model for Turbulent Flows](https://mlsanity.com/paper/1907.09909/) (2019)
- [How to Compute Invariant Manifolds and their Reduced Dynamics in High-Dimensional Finite-Element Models](https://mlsanity.com/paper/2103.10264/) (2021)
- [Principal component trajectories for modeling spectrally-continuous dynamics as forced linear systems](https://mlsanity.com/paper/2005.14321/) (2020)
- [Exact Nonlinear Model Reduction for a von Karman beam: Slow-Fast Decomposition and Spectral Submanifolds](https://mlsanity.com/paper/1703.03001/) (2017)
- [Data-Driven Theory-guided Learning of Partial Differential Equations using SimultaNeous Basis Function Approximation and Parameter Estimation (SNAPE)](https://mlsanity.com/paper/2109.07471/) (2021)

## Built on by (most cited first, 3 of 3)

- [Data-Driven Modeling and Prediction of Non-Linearizable Dynamics via Spectral Submanifolds](https://mlsanity.com/paper/2201.04976/) (2022)
- [VeMo: A Lightweight Data-Driven Approach to Model Vehicle Dynamics](https://mlsanity.com/paper/2510.07447/) (2025)
- [A neural operator for predicting vibration frequency response curves from limited data](https://mlsanity.com/paper/2603.10149/) (2026)

## Abstract (arXiv)

While data-driven model reduction techniques are well-established for linearizable mechanical systems, general approaches to reducing non-linearizable systems with multiple coexisting steady states have been unavailable. In this paper, we review such a data-driven nonlinear model reduction methodology based on spectral submanifolds (SSMs). As input, this approach takes observations of unforced nonlinear oscillations to construct normal forms of the dynamics reduced to very low dimensional invariant manifolds. These normal forms capture amplitude-dependent properties and are accurate enough to provide predictions for non-linearizable system response under the additions of external forcing. We illustrate these results on examples from structural vibrations, featuring both synthetic and experimental data.