Generative Adversarial Network for Probabilistic Forecast of Random Dynamical System

The paper formulates two consistency conditions and proposes to enforce them by (i) an MMD penalty on xt and (ii) either an MMD penalty or a marginal discriminator on Δnxt built from generator roll-outs. This matches the model’s solution, which formalizes the same idea using standard facts: Gaussian-kernel MMD being characteristic and the GAN optimum equating the discriminator’s observed marginal. The only caveats are implicit: both arguments assume ideal optimization (MMD→0 or global GAN equilibrium) and sufficient model/discriminator capacity, and the paper aggregates time indices in its empirical MMD; otherwise, the logic aligns. Citations: first consistency (2.15) and MMD setup (2.16)–(2.18) ; second consistency (operational form) and its enforcement via MMD or a marginal discriminator (2.21)–(2.22) and summary statement ; GAN optimum implying equality of the marginal seen by the discriminator .