{
 "artifacts": [
  {
   "name": "bench_felr.py",
   "url": "https://synthcore.org/code/1077/bench_felr.py"
  },
  {
   "name": "bench_report.json",
   "url": "https://synthcore.org/code/1077/bench_report.json"
  },
  {
   "name": "finite_excitation_replay.py",
   "url": "https://synthcore.org/code/1077/finite_excitation_replay.py"
  },
  {
   "name": "report.md",
   "url": "https://synthcore.org/code/1077/report.md"
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   "name": "report_bench_2026-09-01T235855.md",
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   "name": "results.json",
   "url": "https://synthcore.org/code/1077/results.json"
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 ],
 "category": "training",
 "description": "Replace derivative-based latent-dynamics fitting with an integral regression and maintain a history stack selected by the smallest eigenvalue of its information matrix. The model should perform aggressive parameter updates only when the estimated latent regressors are sufficiently exciting, while a perturbation bound prevents false excitation caused by inaccurate hidden-state estimates.",
 "download_zip": "https://synthcore.org/download/1077",
 "formulas_latex": [
  "$$x(t_i+T)-x(t_i)=\\int_{t_i}^{t_i+T}[Ax(t)+Bu(t)+\\phi(x(t),u(t))]dt+\\Omega_i\\theta+r_i,\\qquad \\Omega_i=\\int_{t_i}^{t_i+T}\\Phi(x(t),u(t))dt.$$",
  "$$\\widehat{G}=\\sum_{i=1}^{M}\\widehat{\\Omega}_i^{\\mathsf T}\\widehat{\\Omega}_i,\\qquad G=\\sum_{i=1}^{M}\\Omega_i^{\\mathsf T}\\Omega_i,$$",
  "$$\\|\\widehat{G}-G\\|\\leq\\sum_{i=1}^{M}\\left(2\\|\\Omega_i\\|\\|\\Delta_i\\|+\\|\\Delta_i\\|^2\\right),\\qquad \\Delta_i=\\widehat{\\Omega}_i-\\Omega_i,$$",
  "$$\\lambda_{\\min}(G)\\geq\\lambda_{\\min}(\\widehat{G})-\\|\\widehat{G}-G\\|.$$"
 ],
 "id": 2914,
 "implementation": "1. Integration point: use a latent ODE, state-space model, or recurrent model with a dynamics head that is linear in an adapter parameter vector \\(\\theta\\); a neural feature map \\(\\Phi_\\psi(z,u)\\) can generate the regressors. Maintain a buffer of windows \\((z(t_i),z(t_i+T),u_{i:i+T})\\), where \\(z\\) is the encoder or recurrent hidden state. 2. Pseudocode: for every stored window, numerically integrate the feature matrix using \\(\\widehat{\\Omega}_i=\\sum_k\\Phi_\\psi(z_k,u_k)\\Delta t\\); form \\(\\widehat G=\\sum_i\\widehat\\Omega_i^{\\mathsf T}\\widehat\\Omega_i\\); estimate latent uncertainty \\(\\varepsilon_i\\) from an ensemble variance, dropout variance, or observer residual; compute \\(q=\\lambda_{\\min}(\\widehat G)-\\sum_i(2\\|\\widehat\\Omega_i\\|\\varepsilon_i+\\varepsilon_i^2)\\). If \\(q\u003e\\gamma\\), update \\(\\theta\\) using the integral-regression residual; otherwise freeze or downweight the dynamics-adapter update. Select history windows greedily by maximizing the increase in \\(\\lambda_{\\min}(\\widehat G)\\). 3. The integral regression and perturbation inequality are taken from the paper; the true matrix and exact state error are unavailable, so \\(\\varepsilon_i\\) must be estimated empirically and inflated by a safety factor. 4. First cheap experiment: train a two- or four-dimensional latent state-space model on Lorenz-63 or a damped nonlinear oscillator, comparing Adam, ungated integral regression, and excitation-gated replay. Sweep observation noise and window length. The quantitative prediction is that parameter error decreases rapidly after \\(q\\) becomes positive, with the empirical transition from ill-conditioned to stable identification near \\(q=0\\). When true states are available for evaluation, measured \\(\\lambda_{\\min}(G)\\) should exceed the conservative lower bound \\(q\\), up to numerical integration error.",
 "math_summary": "Consider a latent state model \\(\\dot{x}=Ax+Bu+\\phi(x,u)+\\Phi(x,u)\\theta+Dd(t)\\), where \\(x\\) is the latent state, \\(u\\) is the input, \\(\\theta\\) is an unknown parameter vector, and \\(d\\) is a bounded disturbance. Over an interval \\([t_i,t_i+T]\\), integration gives \\(x(t_i+T)-x(t_i)=\\int_{t_i}^{t_i+T}[Ax+Bu+\\phi(x,u)]dt+\\Omega_i\\theta+r_i\\), where \\(\\Omega_i=\\int_{t_i}^{t_i+T}\\Phi(x(t),u(t))dt\\) and \\(r_i\\) contains disturbance and modeling residuals. Using an estimated state produces \\(\\widehat{\\Omega}_i=\\Omega_i+\\Delta_i\\). The true information matrix is \\(G=\\sum_i\\Omega_i^{\\mathsf T}\\Omega_i\\), and the computable matrix is \\(\\widehat{G}=\\sum_i\\widehat{\\Omega}_i^{\\mathsf T}\\widehat{\\Omega}_i\\). If \\(\\|\\Delta_i\\|\\leq\\varepsilon_i\\), then \\(\\|\\widehat{G}-G\\|\\leq\\sum_i(2\\|\\Omega_i\\|\\varepsilon_i+\\varepsilon_i^2)\\). Thus \\(\\lambda_{\\min}(G)\\geq\\lambda_{\\min}(\\widehat{G})-\\|\\widehat{G}-G\\|\\), so positive corrected excitation certifies informative data.",
 "math_tags": [
  "control-theory",
  "linear-algebra",
  "spectral-theory",
  "dynamical-systems"
 ],
 "ml_areas": [
  "optimizer",
  "ssm",
  "rnn",
  "world-model"
 ],
 "paper": {
  "arxiv_id": "2608.30977",
  "arxiv_url": "https://arxiv.org/abs/2608.30977",
  "summary_what_math_gives_to_ml": "The paper provides a transferable mechanism for learning unknown dynamics from estimated latent states without persistent excitation: integrate the dynamics to form an output-integral regression, store a finite history stack, and certify that estimated-state excitation remains informative despite observer error and disturbances. Its key quantitative tool is a perturbation bound between true and estimated information matrices, allowing finite excitation to be checked using only computable regressors. A second transferable asset is an OSL-QIB Lyapunov/LMI condition for designing an observer whose latent-state error remains bounded while parameters adapt. These mechanisms suggest stable latent-state sequence models with excitation-aware replay and explicit contraction certificates.",
  "title": "Adaptive Observer of Nonlinear One-Sided Lipschitz Systems Using Estimated State Regressors With Finite Excitation",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 5,
  "novelty": 7,
  "usefulness": 8
 },
 "solves": [
  "stability",
  "sample-efficiency",
  "accuracy"
 ],
 "title": "Finite-Excitation Latent Replay",
 "url": "https://synthcore.org/idea/2914/finite-excitation-latent-replay",
 "verification": {
  "benchmark": {
   "track": "dynamics",
   "model": "rnn_small",
   "n_seeds": 8,
   "metric_direction": "lower is better",
   "comparison": {
    "baseline_mean": 0.0002390248,
    "idea_mean": 0.732533522,
    "delta_mean": 0.7322944972,
    "per_seed_diffs": [
     0.619228,
     0.8258078,
     0.7260383,
     0.8281387,
     0.6346212,
     0.7559647,
     0.6810025,
     0.7875548
    ],
    "idea_wins": 0,
    "n_pairs": 8,
    "p_value": 0.0081,
    "mde": 0.06817817,
    "mde_rel_pct": 28523.48,
    "verdict": "idea worse (significant)",
    "system_worked": false
   },
   "baseline": {
    "best_cfg": {
     "lr": 0.01,
     "weight_decay": 0
    },
    "sweep": [
     {
      "cfg": {
       "lr": 0.001,
       "weight_decay": 0
      },
      "mean": 0.0012379241
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     {
      "cfg": {
       "lr": 0.001,
       "weight_decay": 0.0001
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       "lr": 0.003,
       "weight_decay": 0
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      "mean": 0.0008770637
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      "cfg": {
       "lr": 0.003,
       "weight_decay": 0.0001
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     {
      "cfg": {
       "lr": 0.01,
       "weight_decay": 0
      },
      "mean": 0.0003551548
     },
     {
      "cfg": {
       "lr": 0.01,
       "weight_decay": 0.0001
      },
      "mean": 0.0004823483
     }
    ],
    "full": {
     "mean": 0.0002390248,
     "std": 0.0001635689,
     "per_seed": [
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     "n": 8
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   "idea": {
    "mean": 0.732533522,
    "std": 0.0762778934,
    "per_seed": [
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     0.6348818541,
     0.7560707927,
     0.6810366511,
     0.7876455784
    ],
    "n": 8
   },
   "mechanism_signature": {
    "confirmed": false,
    "predicted_vs_observed": {
     "activation_epoch_values": [
      null,
      null,
      null,
      null,
      null,
      null,
      null,
      null
     ],
     "observed_activation_rate_mean": 0,
     "observed_positive_q_rate_mean": 0,
     "observed_post_activation_loss_drop_mean": 0,
     "predicted_positive_q_activation": true
    },
    "prediction": "parameter updates become active after q exceeds gamma and loss then decreases"
   },
   "custom_track": null
  },
  "mechanism_confirmed": false,
  "peer_reviewed": false,
  "practical_verdict": "harms",
  "stage1_mechanism_check": {
   "worked": true,
   "confidence": 8,
   "verdict": "Built a NumPy MVP implementing integral regressors, Gramian excitation, perturbation correction, and greedy replay selection in finite_excitation_replay.py. The mechanism checks passed: 0/12 perturbation-bound violations, expected small-error O(epsilon) scaling, and a certificate threshold crossing at epsilon 0.240 observed versus 0.242 interpolated. Replay activated at step 4, but its final parameter error was worse than using all data (0.0541 versus 0.00697), so the mechanism is supported while this toy replay policy did not show an identification win.",
   "metrics": {
    "baseline": "All-candidate least squares: final parameter error 0.00697; best error 0.00417.",
    "idea": "Greedy excitation replay: stack size 8, lambda_min=0.6195, activation step 4, final parameter error 0.0541; best error 0.0287. Bound violations: 0/12; certificate threshold epsilon observed/interpolated: 0.240/0.242."
   },
   "how_to_run": "python3 finite_excitation_replay.py",
   "files": [
    "finite_excitation_replay.py",
    "results.json"
   ],
   "limitations": "This is a deterministic linear toy identification test, not a neural latent ODE or Lorenz experiment. It uses one random seed, synthetic matrix regressors, known perturbation magnitudes for validation, and a simple sequential greedy stack rather than a full online observer or ensemble uncertainty estimator; runtime, FLOPs, noisy latent-state estimation, and multi-seed statistical significance were not tested."
  },
  "status": "failed_benchmark",
  "status_label": "Failed on benchmark",
  "updated_at": "2026-09-01T23:58:55",
  "verdict_source": "deterministic test code (paired-seed permutation statistics)"
 }
}