Learning the temporal evolution of multivariate densities via normalizing flows

The paper defines a time-dependent RealNVP coupling layer (its Eqns. (8)–(10)) and uses the time-parameterized change-of-variables formula px(x, t) = pz(T(x, t)) |det JT(x, t)|, justified by augmenting the Jacobian with t so that det[[∂z/∂x, ∂z/∂t]; [0, 1]] = det(∂z/∂x) (paper Eq. (5)) . It also states that the RealNVP block yields det JT = e^{µ(x1:d, t)} and that minimizing the summed negative log-likelihood over time snapshots (Eq. (10)) trains the model and supports sampling via x = T^{-1}(z, t) . The candidate’s solution explicitly derives the block-triangular Jacobian, its determinant, an explicit inverse and C^1 conditions; then shows composition, multiplicativity of determinants, and identifies the objective as maximum likelihood, exactly mirroring the paper’s logic. Their discussion of the non-local Fokker–Planck example aligns with the paper’s treatment and experiments (see the nonlocal PDE and results section) . Thus both are correct and substantively the same proof, with the model supplying a few routine details (block Jacobian and regularity) that the paper states informally.