Data-driven Modeling of Granular Chains with Modern Koopman Theory

The paper defines the deep Koopman architecture and the three MSE losses L_recon, L_lin, and L_pred (its Eq. (23)) and writes the latent relation φ(x_{t+1}) = K φ(x_t) (its Eq. (22)), then asserts (footnote 8) that the learned φ network represents Koopman eigenfunctions when K is diagonalized, but gives no proof or explicit assumptions about injectivity/invertibility of the encoder/decoder or the zero-loss regime. The candidate solution supplies a clean, correct derivation under explicit hypotheses: H1 (φ injective/decoder left-inverse), H2 (K diagonalizable), H3 (losses vanish), thereby showing φ(x_t) = K^t φ(x_0), that the coordinates in K’s eigenbasis are Koopman eigenfunctions (on the training states, and globally if the zero-loss condition holds for all initial states), and that decoded predictions are exact with zero RMSE (matching the paper’s RMSE definition, Eq. (24)).