The Influence of the Memory Capacity of Neural DDEs on the Universal Approximation Property

The paper’s main theorem (Theorem 5.4) precisely states and proves that non-augmented neural DDEs with sufficiently small memory capacity Kτ cannot uniformly approximate a map whose i-th component has a non-degenerate local extremum; the proof uses a careful small-delay decomposition y(·) via G on [0,βτ] and H on [βτ,T], with quantitative bounds (G near the identity on O(τ), and exponential attraction toward special solutions with δ3,τ=C2 e^{−T/τ}), plus a Jordan–Brouwer-based level-set separation that yields a hard geometric contradiction (Sections 5.3–5.5) . By contrast, the candidate solution hinges on a global “closeness-to-identity” estimate for S_T (Step 2) claiming ∥S_T(c1)−S_T(c2)−(c1−c2)∥ ≤ η∥c1−c2∥ with η=(Kτe)/(1−Kτe). This is not established in the paper and is generally false (it contradicts the τ→0 ODE limit, where ∥S_T(c1)−S_T(c2)∥ can scale like e^{KT}∥c1−c2∥, not like ∥c1−c2∥). The paper instead relies on existence/uniqueness and exponential attraction of special solutions for Kτe<1 (Theorem 4.5) and a Lipschitz estimate for I, not a global near-identity property of S_T . Because the candidate’s Step 2 is the linchpin for its extremum and Lipschitz arguments, the model’s proof collapses, whereas the paper’s argument remains coherent and complete.