{
 "artifacts": [
  {
   "name": "experiment.py",
   "url": "https://synthcore.org/code/1213/experiment.py"
  },
  {
   "name": "report.md",
   "url": "https://synthcore.org/code/1213/report.md"
  },
  {
   "name": "results.json",
   "url": "https://synthcore.org/code/1213/results.json"
  }
 ],
 "category": "regularization",
 "description": "Attach a learned nonnegative storage function to a neural state-space model and penalize violations of a strict dissipativity inequality during rollout training. The resulting telescoping inequality limits cumulative output deviation and provides a monitor for whether long-horizon simulations are entering a stable turnpike regime.",
 "download_zip": "https://synthcore.org/download/1213",
 "formulas_latex": [
  "$$ℓ_k−ℓ^*≥V_φ(x_{k+1})−V_φ(x_k)+α(d(y_k,y_k^*))$$",
  "$$Σ_{k=0}^{N-1}α(d(y_k,y_k^*))≤Σ_{k=0}^{N-1}(ℓ_k−ℓ^*)+V_φ(x_0)−V_φ(x_N)$$",
  "$$L_diss=(1/(BN))Σ_{b=1}^{B}Σ_{k=0}^{N-1}[V_φ(x_{k+1})−V_φ(x_k)+ρd_k²−ℓ_k+ℓ_hat^*]_+²$$",
  "$$V_φ(x)=||g_φ(x)||²+δ||x||²,\\quad δ\u003e0,\\quad α(r)=ρr²,\\quad ρ\u003e0$$"
 ],
 "id": 3038,
 "implementation": "Add a small storage network gφ to the neural state-space model. During each rollout, compute xₖ₊₁, yₖ, the ordinary prediction loss ℓₖ, and Vφ(xₖ). Add λdiss times Ldiss to the prediction objective. Use Vφ(x) = ||gφ(x)||² + δ||x||² with a small fixed δ to ensure nonnegativity. Set dₖ to the squared output error relative to the measured target, or use a separately optimized free-initial-state reference trajectory when comparing fixed and free solutions. Estimate ℓ_hat* as a stop-gradient exponential moving minimum of per-step losses; alternatively maintain an auxiliary free-initial-state rollout and use its average loss. Backpropagate through fθ and Vφ, but periodically freeze θ and evaluate dissipativity residuals on held-out sequences. The paper supplies the storage-function telescoping argument; the storage network and constants are learned or estimated. First experiment: compare an unconstrained MLP state-space model with the regularized model on noisy Duffing and Lorenz-63 data for one-step and 100-step prediction. Plot residuals, cumulative output error, and fixed/free initialization gaps versus horizon. The prediction is that average residual becomes nonpositive, cumulative squared deviation saturates or grows sublinearly, and the fixed/free gap remains bounded. Persistent positive residuals or linear cumulative growth falsify the proposed certificate.",
 "math_summary": "Let ℓₖ be the output prediction stage loss, ℓ* a reference optimal stage loss, Vφ(x) ≥ 0 a storage function, and d(yₖ,yₖ*) an output distance to a reference trajectory. Strict dissipativity requires ℓₖ − ℓ* ≥ Vφ(xₖ₊₁) − Vφ(xₖ) + α(d(yₖ,yₖ*)), where α(r) \u003e 0 for r \u003e 0. Summing over time telescopes the storage terms and bounds cumulative deviation by accumulated excess loss plus Vφ(x₀) − Vφ(x_N). Parameterize Vφ(x) = ||gφ(x)||² + δ||x||², with δ \u003e 0, and use α(r) = ρr². Penalize the squared positive part of the inequality residual. The storage decrease and positive output penalty are the mechanisms that prevent persistent trajectory deviation.",
 "math_tags": [
  "control-theory",
  "dynamical-systems",
  "optimization",
  "convex-analysis"
 ],
 "ml_areas": [
  "ssm",
  "rnn",
  "world-model",
  "regularization",
  "training-dynamics"
 ],
 "paper": {
  "arxiv_id": "2609.02071",
  "arxiv_url": "https://arxiv.org/abs/2609.02071",
  "summary_what_math_gives_to_ml": "The paper offers a transferable turnpike mechanism for simulation-error training of nonlinear state-space models. Under reachability, incremental output stability, convex stage costs, and a suitable optimality condition, trajectories optimized from a fixed initial state approach the best trajectory obtained when the initial state is free. This supports shorter recurrent training windows and provides a quantitative prediction: if cumulative deviation is bounded independently of horizon, the average fixed-versus-free trajectory gap decays as O(1/N). The equivalence with strict dissipativity and value-function coercivity also suggests a trainable storage-function regularizer for neural state-space models.",
  "title": "Turnpike properties in nonlinear system identification",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 6,
  "novelty": 6,
  "usefulness": 8
 },
 "solves": [
  "stability",
  "accuracy",
  "generalization"
 ],
 "title": "Dissipative Neural State-Space Identification",
 "url": "https://synthcore.org/idea/3038/dissipative-neural-state-space-identification",
 "verification": {
  "peer_reviewed": false,
  "stage1_mechanism_check": {
   "worked": false,
   "confidence": 9,
   "verdict": "Built a compact Duffing neural state-space identification MVP with a nonnegative learned storage function, squared positive-part dissipativity penalty, 100-step rollout evaluation, and a direct telescoping sanity check. The math check passed numerically: maximum telescoping error was 7.1e-15 and the constructed certified bound had slack of about -20. In the fixed-seed mini-experiment, the dissipative model had worse MSE than the unconstrained baseline (0.1619 vs 0.0341 at 100 steps; 0.001406 vs 0.001326 at 10 steps), while 59.8% of residuals remained positive and mean residual was 0.01445, so the promised stability/accuracy effect was not observed.",
   "metrics": {
    "baseline": "Duffing test MSE: 0.001326 at 10 steps, 0.034096 averaged over 100 steps; cumulative squared error 3.4096",
    "idea": "Duffing test MSE: 0.001406 at 10 steps, 0.161861 averaged over 100 steps; cumulative squared error 16.1861; positive residual fraction 0.598; mean residual 0.01445"
   },
   "how_to_run": "python3 experiment.py",
   "files": [
    "experiment.py",
    "results.json"
   ],
   "limitations": "Only a small damped Duffing toy system and one fixed random seed were tested; Lorenz-63, noisy-data sweeps, fixed/free initialization gaps, FLOPs, plots, hyperparameter searches, and longer training were not evaluated. The baseline residual fields are intentionally NaN because no storage certificate is defined for the baseline."
  },
  "status": "mechanism_failed",
  "status_label": "Mechanism failed",
  "updated_at": "2026-09-03T12:03:28",
  "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": false,
    "tested": true
   }
  }
 }
}