A TRAINING-FREE CONDITIONAL DIFFUSION MODEL FOR LEARNING STOCHASTIC DYNAMICAL SYSTEMS

The paper clearly states and sketches all three claims: (A) the closed-form score expression via differentiating the Gaussian smoothing integral (Eq. (3.10)–(3.11)), (B) the reverse-time SDE ↔ reverse ODE marginal equivalence via the reverse Fokker–Planck identity (Eq. (3.21)–(3.23)), and (C) the Monte Carlo score estimator based on kernel localization around x with a convergence claim as ν→0 and M→∞ (referencing Eq. (3.18)–(3.19) in text and Algorithm 3.1) . However, the paper omits the technical hypotheses and proofs needed to make (B)–(C) rigorous and contains minor inaccuracies (e.g., stating ∇q = q∇ log q “holds for the exponential family,” though it actually holds whenever q is differentiable and positive). By contrast, the candidate solution supplies the needed conditions and a standard proof for (C) via LLN and kernel-consistency, and gives a careful PDE calculation for (B). Hence, while the paper’s statements are directionally correct, its theory write-up is incomplete; the model’s solution is correct under explicit, standard assumptions.