# Alignment-Section Floquet Training for Recurrent Dynamics

- ID: 3015
- Canonical URL: https://synthcore.org/idea/3015/alignment-section-floquet-training-for-recurrent-dynamics
- API JSON: https://synthcore.org/api/idea/3015.json
- API Markdown: https://synthcore.org/api/idea/3015.md
- Verification status: unverified
- Source: [arXiv:2609.01585](https://arxiv.org/abs/2609.01585)
- Category: dynamics
- Solves: stability, accuracy, generalization
- ML areas: rnn, ssm, world-model, training-dynamics
- Math tags: dynamical-systems, control-theory, bifurcations, numerical-analysis, linear-algebra
- Ratings: usefulness 7/10; difficulty 6/10; novelty 7/10

## Idea description

Train a recurrent neural network or state-space model using a Poincare-style event loss: identify two consecutive latent alignment events and require the latent position and velocity at the second event to equal a transformed version of the first. Evaluate the Jacobian of this return map and penalize unstable non-neutral Floquet multipliers, producing long-horizon trajectories that are both periodic or symmetry-periodic and locally stable.

## Mathematical statement

Let the learned continuous-time or discretized latent dynamics be $\dot z=f_\theta(z,u)$ or $z_{k+1}=F_\theta(z_k,u_k)$, with latent state $z\in\mathbb{R}^d$. Let $h(z)=0$ define an event section, such as an alignment condition, and let $\tau(z)$ be the first positive event time after leaving the section. The event return map is $P_\theta(z)=S_\alpha\Phi_\theta(\tau(z),z)$, where $\Phi_\theta$ is the model flow and $S_\alpha$ is a known or learned symmetry transformation parameterized by phase or rotation angle $\alpha$. A relative periodic orbit satisfies $P_\theta(z^*)=z^*$. Its local stability is determined by eigenvalues of $DP_\theta(z^*)$ restricted to the section; eigenvalues associated with time shift or continuous symmetry are neutral and should be projected out. The training mechanism is a return residual plus a spectral penalty on the remaining Floquet multipliers: $\rho_{\mathrm{nonneutral}}=\max_i|\lambda_i|<1$ predicts exponential decay of perturbations at successive events.

## Key formulas

- $$m_i\ddot{\mathbf q}_i=\mathcal G\sum_{j\ne i}\frac{m_im_j}{\|\mathbf q_j-\mathbf q_i\|^3}(\mathbf q_j-\mathbf q_i),$$
- $$P_\theta(z)=S_\alpha\Phi_\theta(\tau(z),z),\qquad h(z)=0,\qquad P_\theta(z^*)=z^*,$$
- $$\mathcal L_{\mathrm{return}}=\|S_\alpha\Phi_\theta(\tau_2,z_1)-z_1\|_2^2+\|S_\alpha\Phi_\theta(\tau_2,z_2)-z_2\|_2^2,$$
- $$\mathcal L_{\mathrm{Floquet}}=\big[\max(0,\rho_{\mathrm{nonneutral}}-1+\varepsilon)\big]^2,\qquad \rho_{\mathrm{nonneutral}}=\max_{\lambda_i\notin\mathcal N}|\lambda_i|,$$

## Implementation notes

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%.

## Disclaimer

AI-generated research hypothesis, automatically tested. Not peer-reviewed.