Multi-Hierarchical Surrogate Learning for Structural Dynamical Crash Simulations Using Graph Convolutional Neural Networks

Part (A) is exactly the additive transfer-learning construction stated in the paper: the latent refinement and decoder refinement are given by z_{k}=Ψ_enc,ℓ(x_ℓ)+Ψ^*_{enc,k}(x_k) and x̆_k=U_k^ℓ x̆_ℓ+Ψ^*_{dec,k}(z_k), with the optional learnable upsampler variant Ψ_dec,k(z)=Θ_k^ℓ(Ψ_dec,ℓ(z))+Ψ^*_{dec,k}(z), all defined explicitly in the multi-hierarchical model description (Eq. (12), (13), (15) in the paper) . Part (B) uses the paper’s node-distance error e2 definition and simply relates it to the per-sample MSE on the original discretization; under the stated hypothesis E_k ≤ E_{k+1} − δ_k, the monotonicity claim follows immediately. The paper itself reports empirical monotone improvement across levels and qualitatively argues residual learning, but does not present a formal theorem or the δ_k hypothesis; hence, the model’s proof is correct while the paper’s argument is incomplete on this point .