# Data-driven modelling of autonomous and forced dynamical systems

arXiv: https://arxiv.org/abs/2512.12432 · HTML: https://mlsanity.com/paper/2512.12432/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 55 references that are in the map.
- First posted on arXiv 13 Dec 2025.

## Builds on (most cited first, 12 of 12)

- [Self-Supervised Learning from Images with a Joint-Embedding Predictive Architecture](https://mlsanity.com/paper/2301.08243/) (2023)
- [V-JEPA 2: Self-Supervised Video Models Enable Understanding, Prediction and Planning](https://mlsanity.com/paper/2506.09985/) (2025)
- [Understanding Dimensional Collapse in Contrastive Self-supervised Learning](https://mlsanity.com/paper/2110.09348/) (2021)
- [Deep Neural Network Approximation Theory](https://mlsanity.com/paper/1901.02220/) (2019)
- [On Convergence of Extended Dynamic Mode Decomposition to the Koopman Operator](https://mlsanity.com/paper/1703.04680/) (2017)
- [Residual Dynamic Mode Decomposition: Robust and verified Koopmanism](https://mlsanity.com/paper/2205.09779/) (2022)
- [Data-Driven Modeling and Prediction of Non-Linearizable Dynamics via Spectral Submanifolds](https://mlsanity.com/paper/2201.04976/) (2022)
- [The mpEDMD Algorithm for Data-Driven Computations of Measure-Preserving Dynamical Systems](https://mlsanity.com/paper/2209.02244/) (2022)
- [Learning normal form autoencoders for data-driven discovery of universal,parameter-dependent governing equations](https://mlsanity.com/paper/2106.05102/) (2021)
- [Data-driven reduced order models using invariant foliations, manifolds and autoencoders](https://mlsanity.com/paper/2206.12269/) (2022)
- [Data-Driven Nonlinear Model Reduction to Spectral Submanifolds via Oblique Projection](https://mlsanity.com/paper/2503.21895/) (2025)
- [Machine-learning invariant foliations in forced systems for reduced order modelling](https://mlsanity.com/paper/2403.14514/) (2024)

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

- [Floquet Fibre Geometry and Higher-Order Reduced Coordinates for Off-Manifold Transients near Nonlinear Aeroelastic Flutter](https://mlsanity.com/paper/2609.14674/) (2026)

## Abstract (arXiv)

The paper demonstrates that invariant foliations are accurate, data-efficient and practical tools for data-driven modelling of physical systems. Invariant foliations can be fitted to data that either fill the phase space or cluster about an invariant manifold. Invariant foliations can be fitted to a single trajectory or multiple trajectories. Over and underfitting are eliminated by appropriately choosing a function representation and its hyperparameters, such as polynomial orders. The paper extends invariant foliations to forced and parameter dependent systems. It is assumed that forcing is provided by a volume preserving map, and therefore the forcing can be periodic, quasi-periodic or even chaotic. The method utilises full trajectories, hence it is able to predict long-term dynamics accurately. We take into account if a forced system is reducible to an autonomous system about a steady state, similar to how Floquet theory guarantees reducibility for periodically forced systems. In order to find an invariant manifold, multiple invariant foliations are calculated in the neighbourhood of the invariant manifold. Some of the invariant foliations can be linear, while others nonlinear but only defined in a small neighbourhood of an invariant manifold, which reduces the number of parameters to be identified. An invariant manifold is recovered as the zero level set of one or more of the foliations. To interpret the results, the identified mathematical models are transformed to a canonical form and instantaneous frequency and damping information are calculated.