# How to Compute Invariant Manifolds and their Reduced Dynamics in High-Dimensional Finite-Element Models

arXiv: https://arxiv.org/abs/2103.10264 · HTML: https://mlsanity.com/paper/2103.10264/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Mar 2021.

## Built on by (most cited first, 8 of 8)

- [Data-Driven Modeling and Prediction of Non-Linearizable Dynamics via Spectral Submanifolds](https://mlsanity.com/paper/2201.04976/) (2022)
- [Data-driven Nonlinear Model Reduction to Spectral Submanifolds in Mechanical Systems](https://mlsanity.com/paper/2110.01929/) (2021)
- [Learning effective dynamics from data-driven stochastic systems](https://mlsanity.com/paper/2205.04151/) (2022)
- [Virtual twins of nonlinear vibrating multiphysics microstructures: physics-based versus deep learning-based approaches](https://mlsanity.com/paper/2205.05928/) (2022)
- [Physics-informed reduced-order modelling with equivariant spectral submanifolds](https://mlsanity.com/paper/2608.04239/) (2026)
- [Information Topology](https://mlsanity.com/paper/2210.03850/) (2022)
- [Revealing hidden correlations from complex spatial distributions: Adjacent Correlation Analysis](https://mlsanity.com/paper/2506.05759/) (2025)
- [Nonlinear Model Order Reduction of Dynamical Systems in Process Engineering: Review and Comparison](https://mlsanity.com/paper/2506.12819/) (2025)

## Abstract (arXiv)

Invariant manifolds are important constructs for the quantitative and qualitative understanding of nonlinear phenomena in dynamical systems. In nonlinear damped mechanical systems, for instance, spectral submanifolds have emerged as useful tools for the computation of forced response curves, backbone curves, detached resonance curves (isolas) via exact reduced-order models. For conservative nonlinear mechanical systems, Lyapunov subcenter manifolds and their reduced dynamics provide a way to identify nonlinear amplitude-frequency relationships in the form of conservative backbone curves. Despite these powerful predictions offered by invariant manifolds, their use has largely been limited to low-dimensional academic examples. This is because several challenges render their computation unfeasible for realistic engineering structures described by finite-element models. In this work, we address these computational challenges and develop methods for computing invariant manifolds and their reduced dynamics in very high-dimensional nonlinear systems arising from spatial discretization of the governing partial differential equations. We illustrate our computational algorithms on finite-element models of mechanical structures that range from a simple beam containing tens of degrees of freedom to an aircraft wing containing more than a hundred-thousand degrees of freedom.