Closed-form Continuous-Depth Models

The paper’s Theorem 1 claims an approximate closed-form for the IVP (1), but its derivation silently replaces the Af(I) drive from (1) with A(wτ + f(I)) when applying the integrating-factor trick, which is the only way their inner integral disappears. This altered ODE is made explicit in their “Integral closed-form solution” step and leads to x(t) = (x0 − A) e^{−wτ t − ∫0^t f(I(s)) ds} + A (their Eq. (4)), not the solution of (1) with b(t) = A f(I(t), θ) . Consequently, the paper’s constant-input limit is A (via the altered ODE), whereas the true equilibrium of (1) is A f0/(wτ + f0). The candidate solution derives the correct integrating-factor formula for the Af(I) case (keeping the nested integral), gives the standard piecewise-constant reduction, and a clean convergence argument—precisely what (1) requires. Hence, the paper’s proof is incorrect as stated for (1), while the model solution is correct for the posed IVP.