{
 "artifacts": null,
 "category": "dynamics",
 "description": "Replace an unconstrained latent recurrent update with a canonical pair of latent states, position-like q and momentum-like p, generated by a learned scalar Hamiltonian. Use a convex quadratic baseline plus a gated nonlinear correction, allowing the model to begin near stable linear dynamics and increase expressivity only when supported by data.",
 "formulas_latex": [
  "$$H[\\eta,\\psi]=\\tfrac{1}{2}\\int\\psi\\,G(\\eta)\\psi\\,d\\bm{x}+\\tfrac{1}{2}g\\int\\eta^{2}\\,d\\bm{x}$$",
  "$$\\partial_t\\eta=\\frac{\\delta H}{\\delta\\psi},\\qquad\\partial_t\\psi=-\\frac{\\delta H}{\\delta\\eta}$$",
  "$$H_\\theta(q,p)=\\tfrac12p^\\top M_\\theta(q)p+V_\\theta(q),\\qquad M_\\theta(q)=L_\\theta(q)L_\\theta(q)^\\top+\\mu I$$",
  "$$H_\\theta(q,p)=H_0(q,p)+\\varepsilon H_{\\mathrm{nl},\\theta}(q,p),\\quad H_0=\\tfrac12p^\\top M_0p+\\tfrac12q^\\top K_0q,\\quad \\dot q=\\nabla_pH_\\theta,\\quad \\dot p=-\\nabla_qH_\\theta$$"
 ],
 "id": 162,
 "implementation": "(1) Integration point: use this as the latent transition in a small video world model, audio sequence model, or partially observed dynamical-system predictor. An encoder maps observation x_t to q_t and p_t; the Hamiltonian block predicts the next latent state, and a decoder maps the latent state back to x_t. For the MVP, use diagonal positive M_0 and K_0 and a compact MLP for H_nl rather than a full matrix-valued kinetic operator. (2) Pseudocode: initialize q,p = encoder(x_t); compute H0 = 0.5 * sum(M0 * p^2) + 0.5 * sum(K0 * q^2); compute Hnl = H_net(q,p); set epsilon = sigmoid(eps_raw); define H = H0 + epsilon * Hnl; calculate dq = autograd(H, p) and dp = -autograd(H, q); perform a symplectic-Euler step p_half = p - dt * dp, q_new = q + dt * autograd(H(q,p_half), p_half), then p_new = p_half - dt * autograd(H(q_new,p_half), q_new). Decode q_new and p_new and train with next-observation loss. Add a small penalty on normalized energy drift over a rollout, |H_T-H_0|/(|H_0|+1e-6). (3) The paper's mathematics supplies the canonical equations and the decomposition into a linear baseline plus epsilon-scaled nonlinear terms. Estimate M0 and K0 from a short system-identification run or initialize them as trainable positive diagonal parameters using softplus. Estimate stability empirically from energy drift, rollout error, and the largest observed latent Jacobian singular value. Keep epsilon in [0,1], and optionally warm it from 0 to its learned value over the first several epochs. (4) First cheap experiment: train a 32- or 64-dimensional latent model on Moving-MNIST or simulated spring-mass trajectories, comparing a GRU, an unconstrained neural ODE, and the Hamiltonian transition at equal parameter count and equal one-step loss. Measure 50- and 200-step rollout MSE, normalized energy drift, gradient norms through time, and the fraction of divergent rollouts. Success means lower long-horizon MSE and fewer exploding trajectories at equal one-step accuracy; a useful target is at least 2x lower 200-step error or a materially lower rollout-failure rate than the unconstrained recurrent baseline.",
 "math_summary": "The paper's Zakharov formulation uses the canonical variables surface height eta and surface potential psi, with Hamiltonian H[eta,psi] = 1/2 integral psi G(eta) psi dx + 1/2 g integral eta^2 dx. Here G(eta) is the Dirichlet–Neumann operator: for boundary value psi, it returns the normal derivative of its harmonic extension. The evolution is exactly partial_t eta = functional derivative of H with respect to psi and partial_t psi = minus the functional derivative of H with respect to eta, so one scalar energy generates both components of the vector field. Adapt this to finite-dimensional latent variables q and p in R^d with H_theta(q,p) = 1/2 p^T M_theta(q) p + V_theta(q), where M_theta(q) = L_theta(q)L_theta(q)^T + mu I is positive definite and mu \u003e 0 ensures conditioning. The canonical equations are dot q = gradient_p H_theta and dot p = -gradient_q H_theta. Introduce the paper's nonlinear-strength idea through H_theta = H_0 + epsilon H_nl,theta, where H_0 = 1/2 p^T M_0 p + 1/2 q^T K_0 q and epsilon is constrained to [0,1].",
 "math_tags": [
  "hamiltonian",
  "symplectic",
  "dynamical-systems",
  "functional-analysis"
 ],
 "ml_areas": [
  "world-model",
  "rnn",
  "training-dynamics"
 ],
 "paper": {
  "arxiv_id": "2608.25203",
  "arxiv_url": "https://arxiv.org/abs/2608.25203",
  "summary_what_math_gives_to_ml": "The paper gives a concrete canonical reduction of a high-dimensional dynamical system to paired state variables whose evolution is generated by a scalar Hamiltonian. The transferable asset is the exact antisymmetric coupling between generalized position and momentum: one learned energy function determines both update directions and constrains rollout dynamics. A neural world model or recurrent latent dynamics module can use this construction with a positive learned kinetic operator and a gated nonlinear correction, increasing expressivity without immediately sacrificing long-horizon stability. The paper's nonlinear-strength parameter and emphasis on stable integration provide directly testable controls for comparing constrained and unconstrained latent dynamics.",
  "title": "Hamiltonian Two-Way Coupling of Nonlinear Waves and 3D Flows",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 5,
  "novelty": 5,
  "usefulness": 7
 },
 "solves": [
  "stability",
  "generalization",
  "sample-efficiency"
 ],
 "title": "Canonical Hamiltonian Latent Dynamics",
 "url": "https://synthcore.org/idea/162/canonical-hamiltonian-latent-dynamics",
 "verification": {
  "peer_reviewed": false,
  "status": "unverified",
  "status_label": "Unverified",
  "verdict_source": "deterministic test code (paired-seed permutation statistics)",
  "verification_axes": {
   "benchmark_mechanism": {
    "confirmed": null,
    "tested": false
   },
   "practical_benchmark": {
    "beats_baseline": null,
    "tested": false
   },
   "toy_mechanism_gate": {
    "confirmed": null,
    "tested": false
   }
  }
 }
}