DySLIM: Dynamics Stable Learning by Invariant Measure for Chaotic Systems

The paper defines unconditional and conditional MMD regularizers between µ* and (S^ℓ_θ)_#µ*, and between (S^ℓ)_#µ* and (S^ℓ_θ)_#µ*, respectively, and recalls that with a characteristic kernel, MMD=0 if and only if the two measures coincide (their Eq. 10 and discussion; see also Sriperumbudur et al. 2010 as cited in the paper) . It also notes µ* = S#µ* = (S^ℓ)#µ*, and explains the equivalence between the unconditional and conditional formulations (their Eq. 13 and Appendix formulas (28)–(29)) , with approximate sampling via time-unrolling (their Eq. 11) and an assumption of existence of µ*θ (footnote 7) . The candidate solution applies exactly this characteristic-kernel fact to Aℓ and Bℓ to derive the invariance and rollout-matching implications. Its optional strengthenings (e.g., two consecutive Aℓ vanish implies one-step invariance; coprime exponents plus invertibility; or B1=0) are mathematically valid under the stated assumptions, though they go beyond what the paper states. Hence both are correct; the paper presents the formulations and properties, while the model spells out immediate consequences not explicitly proved in the paper.