# Symplectic Latent Rollouts

- ID: 784
- Canonical URL: https://synthcore.org/idea/784/symplectic-latent-rollouts
- API JSON: https://synthcore.org/api/idea/784.json
- API Markdown: https://synthcore.org/api/idea/784.md
- Verification status: mechanism_failed
- Source: [arXiv:2607.12143](https://arxiv.org/abs/2607.12143)
- Category: dynamics
- Solves: stability, accuracy
- ML areas: world-model, rnn, ssm, training-dynamics
- Math tags: dynamical-systems, geometry, numerical-analysis, linear-algebra
- Ratings: usefulness 8/10; difficulty 5/10; novelty 5/10

## Idea description

Replace the transition function of a latent world model, recurrent state-space model, or neural ODE with a learned Hamiltonian flow. The network predicts a scalar latent Hamiltonian, while a symplectic integrator generates future states, preserving canonical phase-space structure and suppressing artificial long-horizon energy drift.

## Mathematical statement

Let $z=(q,p)\in\mathbb{R}^{2d}$ be a latent state, where $q\in\mathbb{R}^{d}$ are generalized positions and $p\in\mathbb{R}^{d}$ are conjugate momenta. A neural network $H_\theta(q,p,c)$ represents the scalar Hamiltonian, optionally conditioned on context $c$. Define the canonical matrix $J=\begin{bmatrix}0&I_d\\-I_d&0\end{bmatrix}$. The continuous dynamics are $\dot z=J\nabla_zH_\theta(z,c)$, equivalently $\dot q=\partial H_\theta/\partial p$ and $\dot p=-\partial H_\theta/\partial q$. A leapfrog step of size $h$ is $p_{k+1/2}=p_k-(h/2)\nabla_qH_\theta(q_k,p_k)$, $q_{k+1}=q_k+h\nabla_pH_\theta(q_k,p_{k+1/2})$, and $p_{k+1}=p_{k+1/2}-(h/2)\nabla_qH_\theta(q_{k+1},p_{k+1/2})$. The update is symplectic, meaning its Jacobian $D\Phi_h$ satisfies $(D\Phi_h)^TJD\Phi_h=J$, and for a stable smooth Hamiltonian its numerical energy error is expected to remain bounded and oscillatory rather than grow linearly with rollout time.

## Key formulas

- $$\dot{q}=\frac{\partial H_\theta}{\partial p},\qquad \dot{p}=-\frac{\partial H_\theta}{\partial q},\qquad \dot z=J\nabla_zH_\theta(z,c),\quad J=\begin{bmatrix}0&I_d\\-I_d&0\end{bmatrix}.$$
- $$p_{k+1/2}=p_k-\frac{h}{2}\nabla_qH_\theta(q_k,p_k),\quad q_{k+1}=q_k+h\nabla_pH_\theta(q_k,p_{k+1/2}),\quad p_{k+1}=p_{k+1/2}-\frac{h}{2}\nabla_qH_\theta(q_{k+1},p_{k+1/2}).$$
- $$(D\Phi_h(z))^T J D\Phi_h(z)=J,\qquad \Delta H_k=H_\theta(z_k,c)-H_\theta(z_0,c).$$

## Implementation notes

1. Integration point: use this as the transition operator in a latent world model or state-space model. An encoder maps observations $x_t$ to $z_t=(q_t,p_t)$, the Hamiltonian MLP takes $(q,p,c)$, and a decoder maps predicted latent states back to observations. Use two or four leapfrog substeps per observation interval; train with reconstruction loss, one-step latent matching, and multi-step rollout loss. 2. Pseudocode: encode $z_0=E_\phi(x_0)$; for each rollout step compute $g_q,g_p=\nabla_qH_\theta,\nabla_pH_\theta$; set $p\leftarrow p-hg_q/2$, $q\leftarrow q+hg_p$, recompute $g_q$ at the new $(q,p)$, then set $p\leftarrow p-hg_q/2$; decode every predicted $z$. Backpropagate through the integrator and minimize $L=L_{recon}+\lambda_1L_{one-step}+\lambda_TL_{rollout}$. 3. Computed from the mechanism: the Hamiltonian gradients and symplectic update are autodifferentiated quantities; estimate energy drift, Jacobian symplectic error, and prediction error empirically. Physical energy need not be known because $H_\theta$ is a learned diagnostic. 4. First cheap experiment: train a 4- or 8-dimensional latent model on simulated pendulum trajectories or a small N-body dataset, comparing an MLP transition, explicit-Euler Hamiltonian transition, and leapfrog Hamiltonian transition at equal parameter count. Sweep step size $h$ and rollout horizon $T$. The prediction is that leapfrog $|\Delta H_k|$ remains bounded and approximately oscillatory, with typical amplitude scaling as $O(h^2)$ for a smooth system, while explicit Euler exhibits secular energy drift and becomes unstable at a smaller step size. Also verify that $\|(D\Phi_h)^TJD\Phi_h-J\|_F$ is near autodiff and floating-point error for the leapfrog map.

## Verification

- Status: mechanism_failed
- Mechanism evidence: yes
- Mechanism confirmed: no
- Verdict: Built a differentiable learned-Hamiltonian latent rollout module and quantitative harmonic-oscillator verification. For the separable oscillator, leapfrog had roundoff symplectic defect, energy-error slope 2.000 matching O(h^2), and the predicted stability boundary hω=2; it also outperformed Euler in 500-step trajectory MSE. However, the supplied three-stage update applied to a genuinely nonseparable Hamiltonian produced defects growing with step size, so the claimed general learned-Hamiltonian guarantee was not established.

### Mechanism check

- Verdict: Built a differentiable learned-Hamiltonian latent rollout module and quantitative harmonic-oscillator verification. For the separable oscillator, leapfrog had roundoff symplectic defect, energy-error slope 2.000 matching O(h^2), and the predicted stability boundary hω=2; it also outperformed Euler in 500-step trajectory MSE. However, the supplied three-stage update applied to a genuinely nonseparable Hamiltonian produced defects growing with step size, so the claimed general learned-Hamiltonian guarantee was not established.
- Confidence: 9/10
- Limitations: No observation encoder/decoder, learned world-model training, context-conditioned dataset, parameter-matched neural baseline, or GPU experiment was tested. The implementation follows the supplied explicit kick-drift-kick equations; these are symplectic for separable Hamiltonians but not generally for nonseparable H(q,p).

## Artifacts

- [diagnose.py](https://synthcore.org/code/148/diagnose.py)
- [hamiltonian_module.py](https://synthcore.org/code/148/hamiltonian_module.py)
- [module_check.json](https://synthcore.org/code/148/module_check.json)
- [report.md](https://synthcore.org/code/148/report.md)
- [results.json](https://synthcore.org/code/148/results.json)
- [symplectic_rollout.py](https://synthcore.org/code/148/symplectic_rollout.py)
- [test_module.py](https://synthcore.org/code/148/test_module.py)
- [Download all files as ZIP](https://synthcore.org/download/148)

## Disclaimer

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