Deep Learning for Koopman Operator Estimation in Idealized Atmospheric Dynamics

The paper defines the five MSE-based losses L_recon, L_pred, L_lin, L_noise, and L_repl and explains their intended roles, including linear advancement in latent space via Km and consistency/robustness of predicted and encoded states (see the loss definitions and objective in equations (6)–(11) and their prose explanations in 2409.06522.pdf) . The architecture description explicitly passes the latent code through Km to advance the state, consistent with the one-step and iterated use of Km shown in the experiments and figures . The candidate solution formalizes the immediate algebraic consequences of zero MSE losses—namely, exact reconstruction g^{-1}(g(x_k)) = x_k, exact latent linearity g(x_{k+1}) = Km g(x_k) (either directly from L_lin = 0 or indirectly from L_pred = L_noise = L_repl = 0), and the multi-step identities g(x_{k+t}) = Km^t g(x_k) and x_{k+t} = g^{-1}(Km^t g(x_k)). These follow tautologically from the definitions under the idealized assumption that each individual loss vanishes on the relevant pairs in S, and they match the paper’s intended semantics. Thus the model’s proof is a crisp restatement of what the paper implies; both are correct, and the reasoning is essentially the same, with the model making explicit the equalities that the paper uses informally.