An Improved Autoencoder Conjugacy Network to Learn Chaotic Maps

The paper’s architecture explicitly defines Ũ(x) = h^{-1} ◦ L ◦ h with L = φ^{-1} ◦ T ◦ φ, equivalently Ũ(x) = h^{-1} ◦ φ^{-1} ◦ T ◦ φ ◦ h (their eqs. (7)–(9)), so under the stated conjugacies the composed predictor reduces to the same composition the model calls Ũ; if one assumes the target U truly satisfies U = h^{-1} ◦ L ◦ h, then Ũ = U follows immediately, as the candidate solution notes. However, the paper’s printed loss terms feed U(x) into the encoder path: Lrecon = ∥U(x) − h^{-1}(h(U(x)))∥ and Lpred = ∥U(x) − h^{-1}(φ^{-1}(T(φ(h(U(x))))))∥ (their eqs. (10)–(11)). Literally interpreted, the prediction compares U(x) to Ũ applied to U(x), i.e., U(x) − Ũ(U(x)), which does not vanish even when Ũ = U. This appears to be a notational/formula error: the standard and intended definition compares U(x) to Ũ(x), which would vanish in the ideal exact-conjugacy setting the solver analyzes. The paper also describes h and h^{-1} as network-approximated (not exact inverses) and acknowledges that the assumed conjugacy to the logistic map may fail in some examples; thus, it does not claim the losses are zero in practice. Net: the solver’s derivation and clarification are correct under the intended (and usual) definitions, while the paper’s loss formulas are inconsistent as printed and should be revised.