Nonlinear Proper Orthogonal Decomposition for Convection-Dominated Flows

The paper asserts that NLPOD yields a projection-error-free reduced-order representation by operating on a full-rank/near–full-rank POD expansion and decoding POD coefficients from a low-dimensional latent space, but it provides no formal proof and sometimes conflates “full rank” with “almost full rank” (e.g., RIC = 99.9%) in exposition. The candidate solution supplies the missing linear-algebraic justification: an exact orthogonal error decomposition, the condition for exact reconstruction on training snapshots (n ≥ rank(A) and zero AE reconstruction error), and the SVD–RIC tail relation. These results align with the paper’s intent but add the necessary hypotheses and rigor that the paper does not state explicitly (cf. the paper’s RIC definition and near–full-rank usage; the “elimination of subspace projection error” and “projection-error-free” claims are only true under the model’s explicit conditions). See the paper’s definition of SVD/POD and RIC, and its claims about full/near–full rank and projection-error-free NLPOD for context .