Point Classification with Runge-Kutta Networks and Feature Space Augmentation

The paper derives the discrete adjoint recurrences for an s-stage Runge–Kutta discretization, obtaining p[l+1], stage multipliers p[l]_i, and the stationarity conditions (their (3.20)–(3.22)), then introduces PRK coefficients satisfying the symplectic coupling β̃=β, c̃=c, and β_i ã_{i,j} + β̃_j a_{j,i} − β_i β̃_j = 0 to rewrite the adjoint as a PRK step (their (3.24)–(3.26)), and states that under these conditions the direct and indirect approaches commute, citing prior work rather than proving the commuting identity in detail . Earlier, the paper explicitly frames the goal as choosing partitioned RK methods with non-vanishing weights so that discretization and optimality “formally commute” . The candidate solution gives a clean one-step, discrete integration-by-parts proof that directly establishes p = (∂y^+/∂y)^T p^+ and identifies the PRK adjoint, thereby supplying the missing algebraic details. Minor differences: the paper assumes nonzero weights β_i to define stage adjoints from Lagrange multipliers, while the candidate solution omits stating this assumption explicitly. Overall, both are correct; the paper’s argument is briefer and reference-based, while the model provides an explicit proof.