Watch the video explaining invariant foliations here.
The detailed documentation and a case study can be found here.
The key concept behind methods implemented here is invariance. The following methods are implemented both for autonomous and (quasi-) peridocically forced systems.
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Invariant foliations
- from data
- from discrete-time systems
- from differential equations
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Invariant manifolds from
- differential equations
- discrete-time systems (maps)
- cannot be done directly from data (regardless of what others claim), but can be extracted from foliations
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Invariant manifolds from two invariant foliations
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Accurate instantaneous frequency and damping ratio calculations
The package has been used to demonstrate the methods in papers
- R. Szalai. Machine-learning invariant foliations in forced systems for reduced order modelling, preprint, 2024
- R. Szalai. Non-resonant invariant foliations of quasi-periodically forced systems, preprint, 2024
There are two other papers that explain the background of the methods.
- R. Szalai, Data-Driven Reduced Order Models Using Invariant Foliations, Manifolds and Autoencoders, J Nonlinear Sci 33, 75 (2023). link
- R. Szalai, Invariant spectral foliations with applications to model order reduction and synthesis. Nonlinear Dyn 101, 2645–2669 (2020). link
Paper [4] introduced the idea of using invariant foliations for reduced order modelling, paper [3] has shown that only invariant foliations can be used for genuine data-driven reduced order modelling (when we classify all possible methods into: a) autoencoders, b) invariant foliations, c) invariant manifolds, d) equation-free models.
There are four examples. For paper 1:
- A forced traffic dynamics model
- The forced Shaw-Pierre example
For paper 2:
- A geometrically nonlinear two degree-of-freedom oscillator
- A two-mass oscillator
This package makes the previous version obsolete.
