{
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   "url": "https://synthcore.org/code/581/bench_report.json"
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   "name": "bench_run.py",
   "url": "https://synthcore.org/code/581/bench_run.py"
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   "name": "experiment.py",
   "url": "https://synthcore.org/code/581/experiment.py"
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 ],
 "category": "dynamics",
 "description": "Construct a recurrent or state-space neural module whose latent dynamics are initialized from a mechanistic approximation of the target system rather than from an isotropic random matrix. For traffic-like interacting systems, use a graph reservoir with car-following-inspired relative-position and relative-velocity terms, drive it with undersensed observations, and train a linear or low-rank readout. The mechanism preserves nonlinear state encoding while enforcing an echo-state contraction condition so predictions do not depend sensitively on the arbitrary initial latent state.",
 "download_zip": "https://synthcore.org/download/581",
 "formulas_latex": [
  "$$a(v,s,\\Delta v)=a_{\\max}\\left[1-\\left(\\frac{v}{v_0}\\right)^\\delta-\\left(\\frac{s^*}{s}\\right)^2\\right],\\qquad s^*=s_0+vT+\\frac{v\\Delta v}{2\\sqrt{a_{\\max}b}}.$$",
  "$$x_{k+1}=F_\\theta(x_k,u_k)=x_k+h f_\\theta(x_k,u_k),\\qquad \\hat y_k=W_{\\mathrm{out}}[x_k;u_k].$$",
  "$$\\sup_{x,u}\\left\\|\\frac{\\partial F_\\theta(x,u)}{\\partial x}\\right\\|_2\\leq\\rho\u003c1\\ \\Longrightarrow\\ \\|x_k-x'_k\\|_2\\leq\\rho^k\\|x_0-x'_0\\|_2.$$",
  "$$\\mathcal{L}=\\frac{1}{T}\\sum_{k=1}^{T}\\|\\hat y_k-y_k\\|_2^2+\\lambda\\left[\\max\\left(0,\\hat\\rho-\\rho_{\\mathrm{target}}\\right)\\right]^2,\\qquad \\hat\\rho=\\max_{k\\in\\mathcal{B}}\\sigma_{\\max}\\left(\\frac{\\partial F_\\theta}{\\partial x}(x_k,u_k)\\right).$$"
 ],
 "id": 1487,
 "implementation": "1. Integration point: insert the module between an undersensed input encoder and the prediction head. Build a sparse graph reservoir with one latent state per entity or spatial cell. Each edge computes relative state features such as headway, velocity difference, and local density. Initialize the edge update with the IDM acceleration, then add a small trainable MLP residual. Use RK2, RK4, or explicit Euler integration. Start with a frozen reservoir and ridge-regression readout; only later unfreeze the residual vector field. 2. Pseudocode: initialize $x_0=0$; at each time step compute graph features, calculate the mechanistic acceleration and residual update, set $x_{k+1}=x_k+h f_\\theta(x_k,u_k)$, append $z_k=[x_{k+1};u_k]$, and fit $W_{out}=YZ^T(ZZ^T+\\lambda I)^{-1}$. During residual training, optimize the prediction loss plus the Jacobian penalty in the fourth formula. Estimate $\\hat\\rho$ with one or two power iterations using Jacobian-vector products, avoiding construction of the full Jacobian. 3. Computed from the mechanism: the IDM feature structure, interaction graph, contraction inequality, and closed-form linear readout. Estimated empirically: local Jacobian singular values, effective memory time, and actual alignment with the target dynamics. 4. First cheap experiment: use a 5-20 node traffic graph or synthetic coupled oscillators with 20% of states masked, comparing this model with a vanilla ESN, GRU, and graph neural ODE at equal latent width. Sweep Euler step size $h$ and residual scale. The falsifiable signature is a sharp instability boundary when $\\hat\\rho$ crosses one: hidden-state perturbations should decay geometrically with slope $\\log\\hat\\rho$ below the boundary and grow above it. At $\\hat\\rho\\approx0.9$, a perturbation should shrink by approximately $0.9^k$; measured decay rates should agree within 20%. Dynamics matching should reduce validation error and required readout samples relative to a random reservoir, especially at low observation density.",
 "math_summary": "Let $x(t)\\in\\mathbb{R}^{d}$ be the reservoir state, $u(t)$ the observed input, and $\\dot{x}=f_{\\theta}(x,u)$ its continuous-time dynamics. A traffic-inspired interaction edge from vehicle $j$ to vehicle $i$ can use the IDM acceleration $a(v,s,\\Delta v)=a_{\\max}[1-(v/v_0)^\\delta-(s^*/s)^2]$, where $v$ is speed, $s$ is headway, $\\Delta v=v-v_{\\mathrm{leader}}$, $v_0$ is desired speed, $a_{\\max}$ is maximum acceleration, $\\delta$ is an exponent, and $s^*=s_0+vT+v\\Delta v/(2\\sqrt{a_{\\max}b})$ with standstill gap $s_0$, time headway $T$, and comfortable braking parameter $b$. Generalize this to a graph latent vector field $f_{\\theta}$ and read out $\\hat{y}_k=W_{\\mathrm{out}}z_k$, with $z_k=[x_k;u_k]$. For Euler step size $h$, $x_{k+1}=F_{\\theta}(x_k,u_k)=x_k+h f_{\\theta}(x_k,u_k)$. If $\\sup_{x,u}\\|\\partial F_{\\theta}/\\partial x\\|_2\\leq\\rho\u003c1$, then two states driven by the same input satisfy $\\|x_k-x'_k\\|_2\\leq\\rho^k\\|x_0-x'_0\\|_2$, which is an operational echo-state certificate. When a global bound is unavailable, estimate the local largest singular value along sampled trajectories and penalize violations.",
 "math_tags": [
  "dynamical-systems",
  "control-theory",
  "graph-theory",
  "linear-algebra"
 ],
 "ml_areas": [
  "ssm",
  "rnn",
  "graph-nn",
  "world-model"
 ],
 "paper": {
  "arxiv_id": "2607.27371",
  "arxiv_url": "https://arxiv.org/abs/2607.27371",
  "summary_what_math_gives_to_ml": "The paper offers a transferable reservoir-computing mechanism: use a nonlinear physical dynamical system whose intrinsic evolution is matched to the dynamics of the partially observed target, then train only a linear readout. Its key theoretical asset is the echo-state property, which makes the reservoir state asymptotically independent of its initial condition and therefore usable for stable long-horizon prediction. A neural implementation should replace a generic random ESN with a structured interaction reservoir or latent state-space module initialized from the expected target dynamics, while explicitly monitoring a contraction or memory-decay condition. The most useful experiment is to test whether dynamics matching reduces readout sample complexity while preserving a measurable echo-state stability boundary.",
  "title": "Dynamics-matched Physical Reservoir Computing for Undersensed Traffic Prediction",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 6,
  "novelty": 6,
  "usefulness": 7
 },
 "solves": [
  "stability",
  "accuracy",
  "sample-efficiency"
 ],
 "title": "Dynamics-Matched Contractive Reservoir",
 "url": "https://synthcore.org/idea/1487/dynamics-matched-contractive-reservoir",
 "verification": {
  "benchmark": {
   "track": "dynamics",
   "model": "rnn_small",
   "n_seeds": 8,
   "metric_direction": "lower is better",
   "comparison": {
    "baseline_mean": 0.0003733093799382914,
    "idea_mean": 0.0017617238772800192,
    "delta_mean": 0.0013884144973417278,
    "per_seed_diffs": [
     0.0013501860084943473,
     0.0011444756528362632,
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     0.00147182637010701,
     0.0017291371477767825,
     0.001166194153483957
    ],
    "idea_wins": 0,
    "n_pairs": 8,
    "p_value": 0.0081,
    "mde": 0.00017671141495484378,
    "mde_rel_pct": 47.33645186843536,
    "verdict": "idea worse (significant)",
    "system_worked": false
   },
   "baseline": {
    "best_cfg": {
     "epochs": 15,
     "lr": 0.006,
     "weight_decay": 0
    },
    "sweep": [
     {
      "cfg": {
       "epochs": 15,
       "lr": 0.001,
       "weight_decay": 0
      },
      "mean": 0.0011778276093536988
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     {
      "cfg": {
       "epochs": 15,
       "lr": 0.003,
       "weight_decay": 0
      },
      "mean": 0.0007040829586912878
     },
     {
      "cfg": {
       "epochs": 15,
       "lr": 0.006,
       "weight_decay": 0
      },
      "mean": 0.00042879935062956065
     }
    ],
    "full": {
     "mean": 0.0003733093799382914,
     "std": 0.0001422580194548688,
     "per_seed": [
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      0.0004996434436179698
     ],
     "n": 8
    }
   },
   "idea": {
    "mean": 0.0017617238772800192,
    "std": 0.00017026615425158047,
    "per_seed": [
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     0.001442315406166017,
     0.001998485764488578,
     0.0017929950263351202,
     0.0018884261371567845,
     0.00186431675683707,
     0.0016658375971019268
    ],
    "n": 0
   },
   "mechanism_signature": {
    "confirmed": false,
    "observed_perturbation_ratio": 0.07229285687208176,
    "predicted_rho": 0.5131399999999999,
    "prediction": "contractive recurrent perturbations decay"
   },
   "custom_track": null
  },
  "mechanism_confirmed": false,
  "peer_reviewed": false,
  "practical_verdict": "harms",
  "stage1_mechanism_check": {
   "worked": true,
   "confidence": 8,
   "verdict": "Built a small IDM-inspired masked traffic reservoir and exact contraction sanity checks in experiment.py. The contraction boundary matched rho=gamma*Lambda across the sweep, and measured geometric log-slopes matched log(rho) to below 3e-17 absolute error; the near-one point was numerically neutral as expected. The mechanistic reservoir did not improve prediction in the fair masked-input comparison: MSE was 0.005219 versus 0.004857 for the random reservoir, so there is no evidence of an accuracy win in this toy setup.",
   "metrics": {
    "baseline": "Random contractive ESN masked traffic readout MSE 0.0048568324",
    "idea": "IDM/relative-state contractive reservoir masked traffic readout MSE 0.0052190781; relative change -7.46%. Contraction sweep observed ratios exactly matched predicted gamma*Lambda; decay-slope maximum absolute error was 2.8e-17."
   },
   "how_to_run": "python3 experiment.py",
   "files": [
    "experiment.py",
    "results.json",
    "run_output.txt"
   ],
   "limitations": "The verification uses deliberately constructed linear scaled-orthogonal systems, so it validates the stated contraction mathematics rather than establishing a difficult nonlinear global certificate. The traffic task is a tiny synthetic ring with one fixed seed, one reservoir width, one train/test split, no GRU or graph-neural-ODE comparison, no learned residual training, and no FLOP or runtime study."
  },
  "status": "failed_benchmark",
  "status_label": "Failed on benchmark",
  "updated_at": "2026-09-02T12:30:10",
  "verdict_source": "deterministic test code (paired-seed permutation statistics)",
  "verification_axes": {
   "benchmark_mechanism": {
    "confirmed": false,
    "tested": true
   },
   "practical_benchmark": {
    "beats_baseline": false,
    "tested": true,
    "verdict": "harms"
   },
   "toy_mechanism_gate": {
    "confirmed": true,
    "tested": true
   }
  }
 }
}