Learning from Imperfect Data: Robust Inference of Dynamic Systems using Simulation-based Generative Model

The paper’s Proposition 1 proves that if the Wasserstein-1 distance between generated and observed trajectories is zero, then the coordinate-wise means match and the squared deviations equal the difference of noise variances, followed by a corollary stating equality of generated and true signals when variances match; see the formal statement and proof on pages with Proposition 1 and its derivation and corollary (including the zero-mean assumption and the WGAN setup that uses a variance penalty) . The candidate solution follows the same core idea: W1=0 implies equality in distribution, yielding equality of means and a variance identity; with matching noise variances this forces the trajectory components to coincide. Differences are mainly stylistic (measure-theoretic vs variance-decomposition), and both implicitly rely on independence of p^G and e^G and on finite moments. Overall: both correct, substantially the same proof.