Machine-learning invariant foliations in forced systems for reduced
  order modelling
Robert Szalai
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.