Discovery of Slow Variables in a Class of Multiscale Stochastic Systems via Neural Networks

The paper informally argues that once an encoder–decoder N = D ∘ E approximates the slow projection P, encoder level sets must overlap fast fibers, and even states they are Df-dimensional closed submanifolds. However, it does not supply the regularity or identifiability assumptions needed to justify “submanifold” and “dimension Df,” nor does it make explicit when E must be constant along each fiber. The candidate solution repairs these gaps: it (i) derives that level sets of E lie in unique fast fibers from N = D ∘ E = P and the injectivity of P across fibers, (ii) notes that E becomes constant on fibers if D parametrizes S (injectivity), and (iii) invokes the Regular Value (Preimage) Theorem to obtain that level sets are Df-dimensional embedded submanifolds (for regular values) and are closed as preimages of closed sets under a continuous map. This matches the paper’s intended claims but supplies the missing hypotheses and standard theorem. Compare the paper’s sketch in Section 4, which asserts these conclusions without the needed conditions and its definition of slow maps via alignment with fast fibers in Section 3.4 .