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

The PDF states the invariance equation R(U(x,θ),θ)=U(F(x,θ),θ+ω) with anchoring DR(0,θ)=diag(Λi(θ)) and DU(K(θ),θ) given by the selected left bundles (equations (6) and (8) in the paper), and then formulates Theorem 2: under βm<1, disjoint Sacker–Sell intervals, αj≠0 on the selected index set I, and external nonresonances (12) up to degree ⌊ℶI⌋+1, there exists a unique analytic invariant foliation and the reduced map R can be taken polynomial with only internally resonant terms up to degree ⌊ℶI⌋. However, the paper defers the proof to an external companion paper and provides no substantive argument locally (Theorem 2 ends with “Proof. The details can be found in the paper [28].”). The candidate solution gives a coherent parameterization-method proof sketch that derives and solves the homological equations, explains the role of the spectral quotient in bounding the degree of R, and establishes uniqueness via elimination of nonresonant terms—exactly matching the statement of Theorem 2. Hence the paper’s claim is plausibly correct but unproven here, while the model’s argument is correct at the outlined level.