# Data-driven modelling of nonlinear dynamics by barycentric coordinates and memory

arXiv: https://arxiv.org/abs/2112.06742 · HTML: https://mlsanity.com/paper/2112.06742/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 67 references that are in the map.
- First posted on arXiv 13 Dec 2021.

## Builds on (most cited first, 9 of 9)

- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [Data-Driven Forecasting of High-Dimensional Chaotic Systems with Long Short-Term Memory Networks](https://mlsanity.com/paper/1802.07486/) (2018)
- [Data-driven approximation of the Koopman generator: Model reduction, system identification, and control](https://mlsanity.com/paper/1909.10638/) (2019)
- [Learning dynamical systems from data: a simple cross-validation perspective](https://mlsanity.com/paper/2007.05074/) (2020)
- [Learning Stable Koopman Embeddings](https://mlsanity.com/paper/2110.06509/) (2021)
- [Kernel-based Prediction of Non-Markovian Time Series](https://mlsanity.com/paper/2007.04286/) (2020)
- [Takens-inspired neuromorphic processor: a downsizing tool for random recurrent neural networks via feature extraction](https://mlsanity.com/paper/1907.03122/) (2019)
- [Data-driven model reduction, Wiener projections, and the Koopman-Mori-Zwanzig formalism](https://mlsanity.com/paper/1908.07725/) (2019)

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

- [Deep Learning of Chaotic Systems from Partially-Observed Data](https://mlsanity.com/paper/2205.08384/) (2022)

## Abstract (arXiv)

We present a numerical method to model dynamical systems from data. We use the recently introduced method Scalable Probabilistic Approximation (SPA) to project points from a Euclidean space to convex polytopes and represent these projected states of a system in new, lower-dimensional coordinates denoting their position in the polytope. We then introduce a specific nonlinear transformation to construct a model of the dynamics in the polytope and to transform back into the original state space. To overcome the potential loss of information from the projection to a lower-dimensional polytope, we use memory in the sense of the delay-embedding theorem of Takens. By construction, our method produces stable models. We illustrate the capacity of the method to reproduce even chaotic dynamics and attractors with multiple connected components on various examples.