# Principal-bundle gauge-fixed Hamiltonian network

- ID: 2937
- Canonical URL: https://synthcore.org/idea/2937/principal-bundle-gauge-fixed-hamiltonian-network
- API JSON: https://synthcore.org/api/idea/2937.json
- API Markdown: https://synthcore.org/api/idea/2937.md
- Verification status: unverified
- Source: [arXiv:2608.28278](https://arxiv.org/abs/2608.28278)
- Category: architecture
- Solves: stability, sample-efficiency, accuracy
- ML areas: world-model, ssm, regularization
- Math tags: symplectic, geometry, lie-groups, differential-geometry
- Ratings: usefulness 5/10; difficulty 5/10; novelty 8/10

## Idea description

Represent a time-dependent Hamiltonian system on the reduced state $(q,t,p_q)$ rather than on the redundant extended state $(q,t,p_q,p_t)$. A neural Hamiltonian section predicts one canonical representative of each affine cotangent fiber, while an optional symmetry loss enforces consistency under transformations that translate time.

## Mathematical statement

For a configuration fibration $\Pi:Q\to\mathbb{R}$, define $\eta=\Pi^*(dt)$. The vertical bundle is $V\Pi=\{u\in TQ\mid \eta(\tau_Q(u))(u)=0\}$, where $\tau_Q:TQ\to Q$ is the tangent-bundle projection. Dualizing the inclusion $V\Pi\hookrightarrow TQ$ gives $\mu:T^*Q\to V^*\Pi$. This is a principal $\mathbb{R}$-bundle with action $\psi(s,\alpha_q)=\alpha_q+s\eta(q)$, so covectors differing only in the time-dual component represent the same reduced state. In adapted coordinates, $T^*Q$ has $(q^i,t,p_i,p_t)$ and $V^*\Pi$ has $(q^i,t,p_i)$, with $\mu(q^i,t,p_i,p_t)=(q^i,t,p_i)$ and $\psi(s,(q^i,t,p_i,p_t))=(q^i,t,p_i,p_t+s)$. A Hamiltonian section selects one representative per fiber; parameterize it as $h_\theta(q,t,p_q)=(q,t,p_q,-H_\theta(q,t,p_q))$. The corresponding extended Hamiltonian is $F_\theta(q,t,p_q,p_t)=p_t+H_\theta(q,t,p_q)$, whose zero level is the selected section. For a group action that may translate time, the paper assumes $\Pi\circ\phi_g=\Pi+m(g)$, where $m:G\to\mathbb{R}$ is the time-shift character, with infinitesimal version given by a Lie-algebra 1-cocycle $c_\phi:\mathfrak{g}\to\mathbb{R}$. The neural section can be trained to satisfy equivariance under the induced reduced and extended actions.

## Key formulas

- $$V\Pi=\{u\in TQ\mid \eta(\tau_Q(u))(u)=0\},\qquad \eta=\Pi^*(dt).$$
- $$\psi(s,\alpha_q)=\alpha_q+s\eta(q),\qquad \mu(q^i,t,p_i,p_t)=(q^i,t,p_i),\qquad \psi(s,(q^i,t,p_i,p_t))=(q^i,t,p_i,p_t+s).$$
- $$h_\theta(q,t,p_q)=(q,t,p_q,-H_\theta(q,t,p_q)),\qquad F_\theta(q,t,p_q,p_t)=p_t+H_\theta(q,t,p_q).$$
- $$\Pi\circ\phi_g=\Pi+m(g),\qquad L_{\mathrm{sec}}=\left\|h_\theta(\bar\phi_g z)-\widehat\phi_g h_\theta(z)\right\|^2.$$

## Implementation notes

Integrate the construction into the Hamiltonian vector-field module of a neural ODE or latent world model. The model input is the reduced state $z=(q,t,p_q)$; do not expose the arbitrary extended coordinate $p_t$ to the main network. Compute $H_\theta=\mathrm{MLP}_\theta(q,t,p_q)$ and define the section representative by $p_t=-H_\theta$. For canonical coordinates, obtain dynamics by automatic differentiation: $\dot q=\nabla_{p_q}H_\theta$, $\dot p_q=-\nabla_qH_\theta$, and explicitly set $\dot t=1$. If an extended-state interface is needed, reconstruct $F_\theta=p_t+H_\theta$ and initialize samples on the gauge slice $F_\theta=0$. For a known time-translating symmetry $g$, implement the transformed reduced state $\bar\phi_g(z)$ and extended action $\widehat\phi_g$, then add $L_{\mathrm{sec}}=\|h_\theta(\bar\phi_g z)-\widehat\phi_g h_\theta(z)\|^2$ to the trajectory loss. Pseudocode is: `z=(q,t,pq); H=net(z); pt=-H; dq=grad(H,pq); dpq=-grad(H,q); dt=ones; z2=bar_phi(g,z); Lsec=norm(section(z2)-hat_phi(g,section(z)))**2`. The quotient map, $\eta=dt$, and derivatives are analytic; only the symmetry action is supplied by the simulator. First test on one-dimensional and two-dimensional time-dependent harmonic oscillators, comparing against an extended-coordinate Hamiltonian neural ODE with identical parameter count, solver tolerance, training steps, and trajectory budget. Measure one-step error, long-horizon rollout error, sensitivity to arbitrary $p_t$ shifts, and wall-clock cost. Pre-register these mechanism tests: arbitrary shifts $p_t\mapsto p_t+s$ must change reduced predictions by less than $10^{-5}$; the reduced model should reach a fixed rollout error at least 20% sooner in optimization steps; and symmetry-transformed rollouts should have error within 10% of original rollouts after $L_{\mathrm{sec}}$ training. Remove only the quotient parameterization and only the symmetry loss in ablations. Falsify the transfer if discarding $p_t$ produces no reduction in prediction variance or sample complexity, or if the section model worsens long-horizon error by more than 10% across three seeds.

## Disclaimer

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