Alignment-Section Floquet Training for Recurrent Dynamics

1. Integration point: use a GRU, neural ODE, or diagonal state-space model whose latent state is $z_k$ and whose output is predicted from $z_k$. Add the return loss to the ordinary next-step prediction loss. For a discrete model, define an event by a scalar observable $h(z_k)$ changing sign; for a cheap first version use $h(z)=z_1-c$ and linearly interpolate the crossing time between adjacent steps. Let $z_1$ and $z_2$ be two consecutive crossings, estimate the event-to-event map by unrolling the network, and use either a fixed symmetry $S_\alpha$ or learn $\alpha$ as a scalar constrained to $[0,2\pi]$. 2. Pseudocode: initialize $z$; unroll until $h(z)$ crosses zero twice; save $z_1,z_2$; compute $r=S_\alpha z_2-z_1$; add $\|r\|^2$ to the prediction loss; periodically compute Jacobian-vector products of the event map with autodiff; estimate its leading eigenvalues with 5--10 power or Arnoldi iterations; exclude the known phase direction and penalize $\max(0,|\lambda|-1+\varepsilon)^2$. Backpropagate through the unroll for the return residual and update parameters. 3. Computed versus estimated: the paper supplies the event matching and rotated-monodromy mechanism; the neural Jacobian, event time, and Floquet multipliers are estimated numerically from autodiff and finite event interpolation. 4. First experiment: train a 2-layer GRU on noisy trajectories generated by a planar harmonic oscillator or a two-dimensional rotating limit-cycle system, comparing ordinary teacher forcing against prediction loss plus the return and Floquet terms. Sweep the return-loss coefficient and latent spectral radius. The quantitative prediction is that free-running error after $n$ events scales approximately as $\rho_{\mathrm{nonneutral}}^n$; models with measured $\rho_{\mathrm{nonneutral}}<1$ should show geometric error decay, while long-horizon divergence should begin near the empirical boundary $\rho_{\mathrm{nonneutral}}=1$ within roughly 20%.