{
 "artifacts": [
  {
   "name": "bench_experiment.py",
   "url": "https://synthcore.org/code/1070/bench_experiment.py"
  },
  {
   "name": "bench_report.json",
   "url": "https://synthcore.org/code/1070/bench_report.json"
  },
  {
   "name": "experiment.py",
   "url": "https://synthcore.org/code/1070/experiment.py"
  },
  {
   "name": "report.md",
   "url": "https://synthcore.org/code/1070/report.md"
  },
  {
   "name": "report_bench_2026-09-01T234338.md",
   "url": "https://synthcore.org/code/1070/report_bench_2026-09-01T234338.md"
  },
  {
   "name": "results.json",
   "url": "https://synthcore.org/code/1070/results.json"
  }
 ],
 "category": "dynamics",
 "description": "Treat optimization as a forced dynamical system whose state is the parameter velocity and whose input is the minibatch gradient. Permit ordinary momentum updates below a target energy, but smoothly increase damping when optimizer energy exceeds that target. This preserves less-conservative behavior in low-energy regions while imposing dissipative dynamics during potentially divergent excursions.",
 "download_zip": "https://synthcore.org/download/1070",
 "formulas_latex": [
  "$$E(v)=\\frac{1}{2}v^{\\top}Mv,\\qquad q(E)=\\sigma\\!\\left(\\frac{E-E_\\star}{\\tau}\\right),\\qquad \\sigma(z)=\\frac{1}{1+e^{-z}}.$$",
  "$$M\\dot v=-g-q(E)cMv,\\qquad \\dot\\theta=v.$$",
  "$$\\dot E=v^{\\top}M\\dot v=-v^{\\top}g-2c\\,q(E)E.$$",
  "$$|v^{\\top}g|\\leq B\\ \\Longrightarrow\\ \\dot E\\leq B-2cE\\ \\text{when }q(E)\\approx1,\\qquad \\limsup_{t\\to\\infty}E(t)\\leq\\frac{B}{2c}.$$"
 ],
 "id": 2871,
 "implementation": "Integrate this at the momentum-to-parameter-update interface of SGD with momentum, or after Adam bias correction. Maintain a velocity vector \\(v\\) and choose \\(M=I\\) for the first test. For each minibatch, compute the gradient \\(g\\), calculate \\(E=0.5v^\\top Mv\\), calculate \\(q=\\operatorname{sigmoid}((E-E_\\star)/\\tau)\\), and update the velocity with \\(v\\leftarrow v-\\eta M^{-1}g-\\eta qcv\\); then update parameters using \\(\\theta\\leftarrow\\theta+v\\). With Adam, use the preconditioned bias-corrected first moment as the velocity and apply the same multiplicative damping before changing parameters. The energy, gate, gradient work \\(|v^\\top g|\\), and damping contribution are directly computed. The disturbance bound \\(B\\) is estimated online using an exponential moving maximum of \\(|v^\\top g|\\), rather than assumed known. First test a 3-layer MLP on MNIST with momentum SGD, comparing ordinary momentum, gradient clipping, and the gated optimizer over a learning-rate sweep. Log energy and update norms. The prediction is that high-energy motion becomes bounded with plateau near \\(\\hat B/(2c)\\), decreasing approximately as \\(1/c\\); the gate transition should occur near \\(E=E_\\star\\) with width about \\(4\\tau\\). The ungated divergence boundary should be shifted upward, and the measured plateau should agree with \\(\\hat B/(2c)\\) within approximately 20%.",
 "math_summary": "Let \\(\\theta\\in\\mathbb{R}^d\\) be network parameters, \\(v\\) the optimizer velocity, \\(g\\) the minibatch gradient, and \\(M\\succ0\\) a mass or preconditioning matrix. Define storage energy \\(E(v)=\\frac12v^\\top Mv\\). The transferable semi-passivity condition is imposed only for \\(E\u003eE_\\star\\): the energy derivative must contain a dissipative term proportional to \\(E-E_\\star\\). Use the smooth gate \\(q(E)=\\sigma((E-E_\\star)/\\tau)\\), where \\(E_\\star\\) is the activation threshold, \\(\\tau\u003e0\\) is transition width, and \\(\\sigma\\) is the logistic function. For \\(\\dot\\theta=v\\) and \\(M\\dot v=-g-q(E)cMv\\), with damping \\(c\u003e0\\), the energy derivative is \\(\\dot E=-v^\\top g-2cq(E)E\\). If the forcing satisfies \\(|v^\\top g|\\le B\\), then in the active regime \\(\\dot E\\le B-2cE\\), yielding the ultimate bound \\(\\limsup E\\le B/(2c)\\).",
 "math_tags": [
  "control-theory",
  "dynamical-systems",
  "stochastic-processes",
  "optimization"
 ],
 "ml_areas": [
  "optimizer",
  "training-dynamics",
  "scheduler"
 ],
 "paper": {
  "arxiv_id": "2608.30193",
  "arxiv_url": "https://arxiv.org/abs/2608.30193",
  "summary_what_math_gives_to_ml": "The paper offers a nonstandard semi-passivity mechanism: enforce a passive energy inequality only when system energy exceeds a prescribed threshold, while allowing controlled non-passive behavior below that threshold. A smooth time-varying transition avoids discontinuities from hard switching, and the resulting storage energy remains bounded under disturbances. The direct neural-network transfer is an energy-gated optimizer that permits aggressive updates near a useful basin but activates dissipative damping when momentum or update energy becomes excessive. This produces a measurable energy boundary and a predicted bounded-energy plateau.",
  "title": "Robust Semi-passive Velocity Field Control with Boundedness Guarantees for Safe Interaction between Mechanical Systems and Physical Environment",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 4,
  "novelty": 7,
  "usefulness": 8
 },
 "solves": [
  "stability",
  "speedup",
  "generalization"
 ],
 "title": "Semi-Passive Energy-Gated Optimizer",
 "url": "https://synthcore.org/idea/2871/semi-passive-energy-gated-optimizer",
 "verification": {
  "benchmark": {
   "track": "dynamics",
   "model": "rnn_small",
   "n_seeds": 8,
   "metric_direction": "lower is better",
   "comparison": {
    "baseline_mean": 0,
    "idea_mean": 0.00006680865521957458,
    "delta_mean": -0.000015192361161098233,
    "per_seed_diffs": [
     -0.000020697640138678253,
     -0.00006376119563356042,
     -0.000005472655175253749,
     -0.0000023089232854545116,
     8.147762855514884e-7,
     -0.0000052667874115286395,
     0.0000024185901565942913,
     -0.000027265054086456075
    ],
    "idea_wins": 6,
    "n_pairs": 8,
    "p_value": 0.04715,
    "mde": 0.000018590723573088402,
    "mde_rel_pct": 22.671333104927385,
    "verdict": "idea better (significant)",
    "system_worked": true
   },
   "baseline": {
    "best_cfg": {
     "lr": 0.01,
     "weight_decay": 0
    },
    "sweep": null,
    "full": {
     "mean": 0,
     "std": 0,
     "per_seed": null,
     "n": 0
    }
   },
   "idea": {
    "mean": 0.00006680865521957458,
    "std": 0.00005376186017983789,
    "per_seed": [
     0.00018218360492028296,
     0.00009848197805695236,
     0.00003606851532822475,
     0.00003849564745905809,
     0.00005274467548588291,
     0.000021851565179531462,
     0.000052063427574466914,
     0.00005257982775219716
    ],
    "n": 8
   },
   "mechanism_signature": {
    "E_star": 0.01,
    "c": 10,
    "confirmed": true,
    "mean_abs_grad_work": 0.009492,
    "mean_active_fraction": 0.17375,
    "mean_bound_proxy_B_over_2c": 0.001445,
    "mean_damping_energy_term": 0.41718,
    "mean_gate": 0.280572488,
    "tau": 0.005,
    "transition_width_10_to_90": 0.0219722457735
   },
   "custom_track": null
  },
  "mechanism_confirmed": true,
  "peer_reviewed": false,
  "practical_verdict": "helps",
  "stage1_mechanism_check": {
   "worked": true,
   "confidence": 9,
   "verdict": "Built the semi-passive energy-gated optimizer with explicit velocity, energy, logistic damping gate, toy dynamical verification, and a controlled digits MLP benchmark. The mechanism manifested: q(E) transitioned at E*=2 with the predicted 10–90% width, active unforced decay matched slope -2c within 0.08%, and forced active-regime plateaus matched G²/(2c²) essentially exactly. The optimizer did not improve this benchmark: at lr=0.3 momentum reached 98.22% accuracy versus 97.56% gated, although gating reduced maximum energy from 0.521 to 0.389.",
   "metrics": {
    "baseline": "Momentum SGD: lr 0.03/0.1/0.3 accuracy 97.33%/95.78%/98.22%; max energy 0.0073/0.0549/0.5211.",
    "idea": "Gated optimizer: lr 0.03/0.1/0.3 accuracy 97.11%/94.44%/97.56%; max energy 0.0068/0.0313/0.3891. Toy gate width relative error \u003c1e-11; decay relative errors 0.01%-0.08%; forced plateau relative errors \u003c=3.3e-5."
   },
   "how_to_run": "python3 experiment.py",
   "files": [
    "experiment.py",
    "results.json"
   ],
   "limitations": "The benchmark used sklearn digits rather than full MNIST, one fixed seed, three learning rates, and 300 updates. It did not test the proposed online exponential moving maximum estimate of B, Adam integration, larger networks, multiple seeds, or sustained pathological divergence regimes."
  },
  "status": "beats_baseline",
  "status_label": "Beats tuned baseline",
  "updated_at": "2026-09-01T23:43:38",
  "verdict_source": "deterministic test code (paired-seed permutation statistics)"
 }
}