# Machine-learning invariant foliations in forced systems for reduced order modelling

arXiv: https://arxiv.org/abs/2403.14514 · HTML: https://mlsanity.com/paper/2403.14514/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 30 references that are in the map.
- First posted on arXiv 21 Mar 2024.

## Builds on (most cited first, 4 of 4)

- [Coordinate Descent Converges Faster with the Gauss-Southwell Rule Than Random Selection](https://mlsanity.com/paper/1506.00552/) (2015)
- [Discovering Causal Relations and Equations from Data](https://mlsanity.com/paper/2305.13341/) (2023)
- [Data-Driven Modeling and Prediction of Non-Linearizable Dynamics via Spectral Submanifolds](https://mlsanity.com/paper/2201.04976/) (2022)
- [Data-driven reduced order models using invariant foliations, manifolds and autoencoders](https://mlsanity.com/paper/2206.12269/) (2022)

## Built on by (most cited first, 1 of 1)

- [Data-driven modelling of autonomous and forced dynamical systems](https://mlsanity.com/paper/2512.12432/) (2025)

## Abstract (arXiv)

We identify reduced order models (ROM) of forced systems from data using invariant foliations. The forcing can be external, parametric, periodic or quasi-periodic. The process has four steps: 1. identify an approximate invariant torus and the linear dynamics about the torus; 2. identify a globally defined invariant foliation about the torus; 3. identify a local foliation about an invariant manifold that complements the global foliation 4. extract the invariant manifold as the leaf going through the torus and interpret the result. We combine steps 2 and 3, so that we can track the location of the invariant torus and scale the invariance equations appropriately. We highlight some fundamental limitations of invariant manifolds and foliations when fitting them to data, that require further mathematics to resolve.