{
 "artifacts": [
  {
   "name": "REPORT.md",
   "url": "https://synthcore.org/code/1029/REPORT.md"
  },
  {
   "name": "bench_report.json",
   "url": "https://synthcore.org/code/1029/bench_report.json"
  },
  {
   "name": "experiment.py",
   "url": "https://synthcore.org/code/1029/experiment.py"
  },
  {
   "name": "report.md",
   "url": "https://synthcore.org/code/1029/report.md"
  },
  {
   "name": "report_bench_2026-09-01T185404.md",
   "url": "https://synthcore.org/code/1029/report_bench_2026-09-01T185404.md"
  },
  {
   "name": "results.json",
   "url": "https://synthcore.org/code/1029/results.json"
  },
  {
   "name": "stage2_bench.py",
   "url": "https://synthcore.org/code/1029/stage2_bench.py"
  }
 ],
 "category": "training",
 "description": "Train a neural dynamical surrogate not only to reproduce measured trajectories, but also to reproduce the plant's economically optimal decision and objective value. Add a differentiable decision loss obtained by solving the surrogate's inner optimization problem, and reject models that fit observations while producing extra local optima or a shifted optimum.",
 "download_zip": "https://synthcore.org/download/1029",
 "formulas_latex": [
  "$$J_{\\theta}(u)=c(x_{\\theta}(u),u),\\qquad u_{\\theta}^{\\star}=\\arg\\min_{u\\in\\mathcal U}J_{\\theta}(u),$$",
  "$$L(\\theta)=L_{\\mathrm{data}}(\\theta)+\\lambda_u\\|u_{\\theta}^{\\star}-u^{\\star}\\|_{W_u}^{2}+\\lambda_J\\left|J_{\\theta}(u_{\\theta}^{\\star})-J^{\\star}\\right|^{2},$$",
  "$$\\nabla_uJ_{\\theta}(u_{\\theta}^{\\star})=0,\\qquad \\lambda_{\\min}\\!\\left(\\nabla_u^2J_{\\theta}(u_{\\theta}^{\\star})\\right)\u003e0\\ \\text{for a strict local minimum}.$$"
 ],
 "id": 2711,
 "implementation": "Integrate this at the surrogate-model training level. Use an MLP or neural ODE to predict the state trajectory or steady state from decision vector $u$, then compute the economic objective $J_{\\theta}(u)$ using the same differentiable cost and constraint code used by the real optimizer. For every minibatch or every $K$ training steps, solve the inner problem with projected L-BFGS, SQP, or differentiable unrolled gradient descent from several initial points. Pseudocode is: (1) predict $x_{\\theta}(u)$ on measured data and compute $L_{\\mathrm{data}}$; (2) initialize $M$ decision vectors; (3) repeat $u_m\\leftarrow\\Pi_{\\mathcal U}[u_m-\\rho\\nabla_uJ_{\\theta}(u_m)]$ for $T$ steps; (4) select the lowest feasible objective as $u_{\\theta}^{\\star}$; (5) backpropagate $L_{\\mathrm{data}}+\\lambda_u\\|u_{\\theta}^{\\star}-u^\\star\\|^2+\\lambda_J|J_{\\theta}(u_{\\theta}^{\\star})-J^\\star|^2$. The paper supplies the failure mechanism and the decision benchmark; the plant optimum, feasible set, and cost are supplied by the application, while gradients and Hessians are computed by autodiff and the number of optima is estimated by multistart. First experiment: fit a neural ODE on synthetic nonlinear process data where the true objective has one known optimum, compare MSE-only training with the added decision loss, and run 100 multistart optimizations. The quantitative prediction is that MSE-only models will sometimes have several local minima and a nonzero optimum displacement, while decision-aware training should reduce $\\|u_{\\theta}^{\\star}-u^\\star\\|$ and the number of distinct converged optima; measure success as at least a 50% reduction in optimum displacement and no extra minima in the well-conditioned benchmark.",
 "math_summary": "Let $f_{\\theta}$ be a learned process model, $u\\in\\mathcal U$ the vector of operating decisions, $x_{\\theta}(u)$ the predicted steady state or rollout, and $J_{\\theta}(u)=c(x_{\\theta}(u),u)$ the predicted economic objective to minimize. Let $u^\\star$ and $J^\\star$ be the trusted plant optimum from a benchmark or high-fidelity simulator. In addition to trajectory loss $L_{\\mathrm{data}}(\\theta)$, minimize $L_{\\mathrm{dec}}(\\theta)=\\|u_{\\theta}^\\star-u^\\star\\|_{W_u}^2+\\alpha|J_{\\theta}(u_{\\theta}^\\star)-J^\\star|^2$, where $u_{\\theta}^\\star=\\arg\\min_{u\\in\\mathcal U}J_{\\theta}(u)$ and $W_u\\succeq0$ weights decision coordinates. A local optimum must satisfy $\\nabla_uJ_{\\theta}(u)=0$ and $\\nabla_u^2J_{\\theta}(u)\\succeq0$. The mechanism predicts that trajectory-equivalent models can have substantially different $u_{\\theta}^\\star$; decision loss removes this non-identifiability in directions relevant to control.",
 "math_tags": [
  "optimization",
  "dynamical-systems",
  "control-theory",
  "statistics"
 ],
 "ml_areas": [
  "optimizer",
  "loss",
  "world-model",
  "training-dynamics"
 ],
 "paper": {
  "arxiv_id": "2608.23885",
  "arxiv_url": "https://arxiv.org/abs/2608.23885",
  "summary_what_math_gives_to_ml": "The paper identifies a practically important failure mechanism: low prediction error on historical trajectories does not preserve the economically relevant optimum of a dynamical process. Neural surrogate models can create multiple phantom optima, and stochastic gradient training can move from weights that recover the correct optimum to observationally similar weights with poor decision performance. The strongest transfer to neural networks is decision-oriented training and validation: explicitly test and regularize the surrogate's optimal-control landscape rather than relying only on trajectory fit.",
  "title": "A tale of perfect fit and phantom optima: how data-driven models can fail in real-time optimization",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 6,
  "novelty": 7,
  "usefulness": 8
 },
 "solves": [
  "accuracy",
  "stability",
  "generalization"
 ],
 "title": "Decision-Oriented Optimum Preservation",
 "url": "https://synthcore.org/idea/2711/decision-oriented-optimum-preservation",
 "verification": {
  "benchmark": {
   "track": "dynamics",
   "model": "rnn_small",
   "n_seeds": 8,
   "metric_direction": "lower is better",
   "comparison": {
    "baseline_mean": 0.002105819425196387,
    "idea_mean": 0.7633235901594162,
    "delta_mean": 0.7612177707342198,
    "per_seed_diffs": [
     0.7447684307117015,
     0.7350623806705698,
     0.8001017065253109,
     0.8846517286729068,
     0.7334783706705698,
     0.7448768028989434,
     0.7399664083495736,
     0.7068363376893103
    ],
    "idea_wins": 0,
    "n_pairs": 8,
    "p_value": 0.0081,
    "mde": 0.04703323792024009,
    "mde_rel_pct": 2233.4886532758533,
    "verdict": "idea worse (significant)",
    "system_worked": false
   },
   "baseline": {
    "best_cfg": {
     "epochs": 12,
     "lr": 0.006
    },
    "sweep": [
     {
      "cfg": {
       "epochs": 12,
       "lr": 0.001
      },
      "mean": 0.004298779065720737
     },
     {
      "cfg": {
       "epochs": 12,
       "lr": 0.003
      },
      "mean": 0.0034615940821822733
     },
     {
      "cfg": {
       "epochs": 12,
       "lr": 0.006
      },
      "mean": 0.0018282315868418664
     }
    ],
    "full": {
     "mean": 0.002105819425196387,
     "std": 0.000389436566226016,
     "per_seed": [
      0.002004395006224513,
      0.001437882543541491,
      0.0015748648438602686,
      0.002295783953741193,
      0.0024020641576498747,
      0.0024898042902350426,
      0.0020851334556937218,
      0.0025566271506249905
     ],
     "n": 8
    }
   },
   "idea": {
    "mean": 0.7633235901594162,
    "std": 0.05257852286312429,
    "per_seed": [
     0.746772825717926,
     0.7365002632141113,
     0.8016765713691711,
     0.886947512626648,
     0.735880434513092,
     0.7473666071891785,
     0.7420515418052673,
     0.7093929648399353
    ],
    "n": 8
   },
   "mechanism_signature": {
    "confirmed": false,
    "decision_probe": {
     "bounds": [
      -1.5,
      1.5
     ],
     "contexts": 32,
     "multistart": 5
    },
    "measured_on_trained_models": true,
    "note": "Standard task test MSE was used as the primary metric; the decision probe was diagnostic only.",
    "prediction": "decision-aware training reduces surrogate optimum displacement relative to MSE-only while preserving trajectory fit"
   },
   "custom_track": null
  },
  "mechanism_confirmed": false,
  "peer_reviewed": false,
  "practical_verdict": "harms",
  "stage1_mechanism_check": {
   "worked": true,
   "confidence": 8,
   "verdict": "Built a differentiable surrogate with unrolled projected inner optimization and decision loss, plus fixed-seed math sweeps. The quadratic inner-loop stability boundary was confirmed at rho=2/H: predicted 0.5405, with contraction at 0.50 and divergence at 0.55. Objective perturbations produced extra minima as epsilon increased, and decision-aware training reduced optimum displacement from 0.40495 to 0.01505 (96.3%) with nearly unchanged data MSE. The mechanism manifested, but the stronger claim about removing extra minima was not established because the simple multistart gradient-flow count produced flat-region artifacts.",
   "metrics": {
    "baseline": "MSE-only: optimum u=0.160, displacement=0.40495, objective=0.01587, data MSE=0.07961, grid minima=1; multistart endpoint count=19 but unreliable.",
    "idea": "Decision-aware: optimum u=0.580, displacement=0.01505, objective=0.01005, data MSE=0.08007, grid minima=1; multistart endpoint count=14 but unreliable. Stability: predicted boundary rho=0.54054, observed transition between 0.50 stable and 0.55 divergent. Perturbation sweep: 1 minimum through epsilon=0.01, 2 at 0.02, 3 at 0.04."
   },
   "how_to_run": "python3 experiment.py",
   "files": [
    "experiment.py",
    "results.json",
    "REPORT.md"
   ],
   "limitations": "Only a one-dimensional synthetic steady-state surrogate was tested; no neural ODE, constraints beyond box projection, noisy trajectories, multiple decision dimensions, computational/FLOP comparison, or robust Hessian-filtered/local-root basin counting was performed. The simple multistart gradient-flow endpoint counts should not be interpreted as distinct local-minimum counts."
  },
  "status": "failed_benchmark",
  "status_label": "Failed on benchmark",
  "updated_at": "2026-09-01T18:54:04",
  "verdict_source": "deterministic test code (paired-seed permutation statistics)"
 }
}