Generating-Function Symplectic Layer

(1) Integration point: use this as a recurrent or state-space block, replacing a GRU or ordinary second-order recurrence. Split the hidden state into q_t and p_t in R^d, and decode from their concatenation. For the first MVP, use rho_theta(v)=0.5 v^T M v with M=R^T R+epsilon I; later replace it by a learned convex neural potential. (2) Given q_t and q_{t+1}, initialize z^0=2q_{t+1}-q_t. For k=0,...,K-1, compute F(z)=gradient_2 L_theta(q_t,q_{t+1})+gradient_1 L_theta(q_{t+1},z), then update z using damped Newton or gradient descent: z^{k+1}=z^k-eta J_F(z^k)^{-1}F(z^k). Set q_{t+2}=z^K and p_{t+1}=-gradient_1 L_theta(q_t,q_{t+1}). For an input-driven model, solve F(z)-B x_t=0. (3) Compute all derivatives with autodiff. The paper-derived part is the implicit generating-function equation and invariant two-form; solver tolerance, damping, and input forcing are empirical choices. Estimate symplectic error by finite differences or autodiff: form the Jacobian J of the map from (q_t,p_t) to (q_{t+1},p_{t+1}) and measure ||J^T Omega J-Omega||_F, where Omega is the block matrix [[0,I],[-I,0]]. (4) First cheap experiment: train a two-layer 128-dimensional model on adding problems and permuted sequential MNIST, comparing a GRU and an unconstrained second-order recurrence at matched parameter count. Test lengths beyond those used in training. Record validation accuracy, gradient norms, hidden-state norm, symplectic error, and wall-clock cost. A positive result is bounded gradient and state drift with better long-context accuracy at similar one-step loss; also test whether warm-starting the implicit solve keeps overhead below twice the baseline recurrent cost.