{
 "artifacts": [
  {
   "name": "bench_lyapunov.py",
   "url": "https://synthcore.org/code/1132/bench_lyapunov.py"
  },
  {
   "name": "experiment.py",
   "url": "https://synthcore.org/code/1132/experiment.py"
  },
  {
   "name": "report.md",
   "url": "https://synthcore.org/code/1132/report.md"
  },
  {
   "name": "results.json",
   "url": "https://synthcore.org/code/1132/results.json"
  }
 ],
 "category": "dynamics",
 "description": "Replace an unconstrained recurrent or neural-ODE hidden-state evolution with a parameter-conditioned vector field whose Jacobian is contractive in a learned positive-definite metric. A Lyapunov residual is added during training using the current context, time, or operating-condition vector, allowing one model to remain stable across changing regimes rather than only near one nominal point.",
 "download_zip": "https://synthcore.org/download/1132",
 "formulas_latex": [
  "$$\\dot z=f_\\phi(z,\\theta,t),\\qquad A_\\phi(z,\\theta,t)=\\frac{\\partial f_\\phi}{\\partial z},\\qquad V(\\delta z,t)=\\delta z^\\top P_\\psi(\\theta,t)\\delta z.$$",
  "$$\\dot P_\\psi+A_\\phi^\\top P_\\psi+P_\\psi A_\\phi+2\\alpha P_\\psi\\preceq0,\\qquad P_\\psi\\succeq p_{\\min}I\u003e0.$$",
  "$$J_k^\\top P_{k+1}J_k\\preceq e^{-2\\alpha h}P_k,\\qquad J_k=I+hA_k.$$",
  "$$\\mathcal{L}_{\\mathrm{lyap}}=\\frac{1}{N}\\sum_{k=0}^{N-1}\\left[\\max\\left(0,\\lambda_{\\max}\\left(P_k^{-1/2}(J_k^\\top P_{k+1}J_k-e^{-2\\alpha h}P_k)P_k^{-1/2}\\right)\\right)\\right]^2.$$"
 ],
 "id": 2938,
 "implementation": "1. Integration point: use a residual RNN, neural ODE solver, or latent state-space block. Parameterize $P_\\psi(\\theta,t)=L_\\psi(\\theta,t)L_\\psi(\\theta,t)^\\top+p_{\\min}I$, where $L_\\psi$ is lower triangular with positive diagonal. 2. Pseudocode: compute $z_{k+1}=z_k+h f_\\phi(z_k,\\theta_k,t_k)$; obtain $A_k$ explicitly for small states or through Jacobian-vector products and power iteration for large states; evaluate $P_k$ and $P_{k+1}$; compute the largest generalized eigenvalue of $(J_k^\\top P_{k+1}J_k,e^{-2\\alpha h}P_k)$; add its positive part to the task loss; update $\\phi$ and $\\psi$. 3. Computed from the mechanism: the Lyapunov inequality, decay rate $\\alpha$, and metric positivity. Estimated empirically: local Jacobians, parameter derivatives of $P$, and minibatch averages over trajectories. 4. First experiment: train equal-width GRUs and residual MLP dynamics on sequence-copy and Lorenz-63 prediction tasks. Compare vanilla training, spectral normalization, and this penalty at equal optimizer steps; ablate both the Lyapunov term and parameter dependence of $P$. 5. Pre-registered signature: perturbation energy must satisfy $V_k\\le e^{-2\\alpha kh}V_0$ up to discretization error, and the measured contraction factor must cross one near $\\lambda_{\\max}(P_k^{-1/2}J_k^\\top P_{k+1}J_kP_k^{-1/2})=e^{-2\\alpha h}$. Reject the transfer if the empirical boundary differs by more than 20%, or if the Lyapunov residual is positive without reducing perturbation growth versus the ablation.",
 "math_summary": "Let $z(t)\\in\\mathbb{R}^{n}$ be the hidden state, $\\theta(t)$ a measurable conditioning parameter, and $\\dot z=f_\\phi(z,\\theta,t)$. The local generator is $A_\\phi(z,\\theta,t)=\\partial f_\\phi/\\partial z$. Let $P_\\psi(\\theta,t)=P_\\psi(\\theta,t)^\\top\\succ p_{\\min}I$ be a parameter-dependent Lyapunov metric, where $p_{\\min}\u003e0$. The transferred differential LMI is $\\dot P_\\psi+A_\\phi^\\top P_\\psi+P_\\psi A_\\phi+2\\alpha P_\\psi\\preceq0$, with desired decay rate $\\alpha\u003e0$. It implies that $V(\\delta z,t)=\\delta z^\\top P_\\psi\\delta z$ satisfies $\\dot V\\le-2\\alpha V$ for infinitesimal perturbations. For a residual layer $z_{k+1}=z_k+h f_\\phi(z_k,\\theta_k)$, the finite-step condition is $J_k^\\top P_{k+1}J_k\\preceq e^{-2\\alpha h}P_k$, where $J_k=I+hA_k$.",
 "math_tags": [
  "control-theory",
  "dynamical-systems",
  "linear-algebra",
  "optimization"
 ],
 "ml_areas": [
  "rnn",
  "ssm",
  "optimizer",
  "training-dynamics"
 ],
 "paper": {
  "arxiv_id": "2608.28349",
  "arxiv_url": "https://arxiv.org/abs/2608.28349",
  "summary_what_math_gives_to_ml": "The paper provides a transferable robust-stability mechanism: parameter-dependent Lyapunov operators satisfying differential operator LMIs certify uniform exponential stability for non-autonomous evolution systems, while Galerkin projections turn infinite-dimensional inequalities into finite-dimensional synthesis constraints. This can be transferred to parameter-conditioned neural ODEs, state-space models, or recurrent networks by treating hidden-state Jacobians as time- and parameter-dependent generators and learning a positive-definite metric that contracts them. Its H-infinity extension also supplies a concrete disturbance-attenuation certificate for neural observers operating with noisy or partial observations.",
  "title": "Operator-Theoretic Stability and Observer Synthesis for Parameter-Dependent Vlasov--Maxwell Dynamics",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 6,
  "novelty": 6,
  "usefulness": 8
 },
 "solves": [
  "stability",
  "accuracy"
 ],
 "title": "Parameter-Dependent Lyapunov Neural Dynamics",
 "url": "https://synthcore.org/idea/2938/parameter-dependent-lyapunov-neural-dynamics",
 "verification": {
  "peer_reviewed": false,
  "stage1_mechanism_check": {
   "worked": true,
   "confidence": 8,
   "verdict": "Built a compact parameter-conditioned residual dynamics model with a learned positive-definite metric and finite-step generalized-eigenvalue Lyapunov penalty. The exact 2D check satisfied the finite-step LMI (maximum normalized residual eigenvalue -0.083) and the measured energy ratio 0.797 was below the theoretical bound 0.946. In the learned comparison, the penalty reduced violating transitions from 44.5% to 3.9% and reduced the mean residual eigenvalue from 0.634 to -0.247, but prediction MSE worsened from 0.0199 to 0.0233, so the stability effect is real while there was no accuracy win.",
   "metrics": {
    "baseline": "MSE 0.01987; Lyapunov violation fraction 0.445; mean generalized residual eigenvalue 0.634; local contraction ratio mean 0.679",
    "idea": "MSE 0.02332; Lyapunov violation fraction 0.039; mean generalized residual eigenvalue -0.247; local contraction ratio mean 0.681"
   },
   "how_to_run": "python3 experiment.py",
   "files": [
    "experiment.py",
    "results.json"
   ],
   "limitations": "Only a tiny 2D one-step residual-dynamics task was tested for 120 optimization steps. The parameter transition used for P_{k+1} was a rolled minibatch condition rather than a generated multi-step trajectory, and no GRU, Lorenz-63, spectral-normalization baseline, long-horizon robustness test, or equal-FLOP study was run."
  },
  "status": "unverified",
  "status_label": "Unverified",
  "updated_at": "2026-09-02T12:51:45",
  "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": true,
    "tested": true
   }
  }
 }
}