{
 "artifacts": [
  {
   "name": "experiment.py",
   "url": "https://synthcore.org/code/9/experiment.py"
  },
  {
   "name": "report.md",
   "url": "https://synthcore.org/code/9/report.md"
  },
  {
   "name": "results.json",
   "url": "https://synthcore.org/code/9/results.json"
  }
 ],
 "category": "dynamics",
 "description": "Train a learned vector field through the same finite-step integrator used for forecasting, rather than fitting only instantaneous derivatives or one-step transitions. Increase the rollout horizon during optimization and combine multi-step error with sparsity, forcing the model to remain self-consistent under repeated composition and reducing exposure bias.",
 "download_zip": "https://synthcore.org/download/9",
 "formulas_latex": [
  "$$\\frac{d\\bm{x}}{dt}=\\bm{f}(\\bm{x}),\\qquad \\bm{x}\\in\\mathbb{R}^{n},\\quad \\bm{f}:\\mathbb{R}^{n}\\to\\mathbb{R}^{n}.$$",
  "$$\\tilde{\\bm{x}}[h+1]=\\tilde{\\bm{x}}[h]+\\frac{1}{6}(\\bm{k}_{1}+2\\bm{k}_{2}+2\\bm{k}_{3}+\\bm{k}_{4})\\equiv\\tilde{\\bm{S}}(\\tilde{\\bm{x}}[h]),$$",
  "$$\\bm{k}_{1}=\\Delta t\\tilde{\\bm{f}}(\\tilde{\\bm{x}}[h]),\\quad \\bm{k}_{2}=\\Delta t\\tilde{\\bm{f}}(\\tilde{\\bm{x}}[h]+\\bm{k}_{1}/2),\\quad \\bm{k}_{3}=\\Delta t\\tilde{\\bm{f}}(\\tilde{\\bm{x}}[h]+\\bm{k}_{2}/2),\\quad \\bm{k}_{4}=\\Delta t\\tilde{\\bm{f}}(\\tilde{\\bm{x}}[h]+\\bm{k}_{3}).$$",
  "$$\\mathcal{L}_{H}(\\theta)=\\frac{1}{H}\\sum_{j=1}^{H}\\left\\|\\tilde{\\bm{x}}[j]-\\bm{x}[j]\\right\\|_{1}+\\lambda\\|\\theta\\|_{1},\\qquad \\tilde{\\bm{x}}[0]=\\bm{x}[0].$$"
 ],
 "id": 44,
 "implementation": "Integrate this at the training loss of a learned dynamics or world-model module, replacing teacher-forced one-step prediction with differentiable RK4 rollouts. Inputs are a batch of trajectories $x_{b,0:H_{\\max}}$ and a vector-field network $f_{\\theta}$. For each curriculum phase, choose horizon $H$, set $\\hat{x}_{b,0}=x_{b,0}$, and execute the following loop: compute $k_1=dt f_\\theta(\\hat{x}_{b,j})$; compute $k_2=dt f_\\theta(\\hat{x}_{b,j}+k_1/2)$; compute $k_3=dt f_\\theta(\\hat{x}_{b,j}+k_2/2)$; compute $k_4=dt f_\\theta(\\hat{x}_{b,j}+k_3)$; then set $\\hat{x}_{b,j+1}=\\hat{x}_{b,j}+(k_1+2k_2+2k_3+k_4)/6$. Accumulate the mean absolute error against $x_{b,j+1}$ and add $\\lambda\\|\\theta\\|_1$. Start with $H=1$ for several epochs, then increase geometrically, for example $1,2,4,8,16$, while optionally normalizing coordinates and clipping gradients. The RK4 formula and loss are taken from the paper; the schedule, gradient clipping, and normalization are engineering choices. Measure rollout error at horizons longer than training, maximum state norm, gradient explosions, and long-time means and variances. The first cheap test is a 2D pendulum, Van der Pol, and Lorenz system with 1% additive noise, comparing one-step MSE training, fixed long-horizon training, and the curriculum at equal vector-field evaluations. Success means lower 32- or 64-step error, fewer divergent trajectories, and improved recovery of long-time statistics at equal training FLOPs.",
 "math_summary": "The paper models an autonomous system as $d\\bm{x}/dt=\\bm{f}(\\bm{x})$, where $\\bm{x}\\in\\mathbb{R}^{n}$ is the state and $\\bm{f}:\\mathbb{R}^{n}\\to\\mathbb{R}^{n}$ is the vector field. Its explicit fourth-order Runge–Kutta step defines a discrete map $\\tilde{\\bm{S}}$ by $\\tilde{\\bm{x}}[h+1]=\\tilde{\\bm{x}}[h]+(\\bm{k}_{1}+2\\bm{k}_{2}+2\\bm{k}_{3}+\\bm{k}_{4})/6$, with $\\bm{k}_{1}=\\Delta t\\tilde{\\bm{f}}(\\tilde{\\bm{x}}[h])$, $\\bm{k}_{2}=\\Delta t\\tilde{\\bm{f}}(\\tilde{\\bm{x}}[h]+\\bm{k}_{1}/2)$, $\\bm{k}_{3}=\\Delta t\\tilde{\\bm{f}}(\\tilde{\\bm{x}}[h]+\\bm{k}_{2}/2)$, and $\\bm{k}_{4}=\\Delta t\\tilde{\\bm{f}}(\\tilde{\\bm{x}}[h]+\\bm{k}_{3})$. Here $h$ is the discrete time index, $\\Delta t$ is the sampling interval, $\\tilde{\\bm{f}}$ is the learned vector field, and $\\tilde{\\bm{S}}$ is the learned one-step transition. Use the adapted objective $\\mathcal{L}_{H}=\\frac{1}{H}\\sum_{j=1}^{H}\\|\\tilde{\\bm{x}}[j]-\\bm{x}[j]\\|_{1}+\\lambda\\|\\theta\\|_{1}$, where $H$ is the rollout horizon, $\\theta$ contains vector-field parameters, and $\\lambda$ controls sparsity. The horizon schedule $H=1\\rightarrow H_{\\max}$ first provides stable gradients and then penalizes errors amplified by repeated composition.",
 "math_tags": [
  "dynamical-systems",
  "numerical-analysis",
  "optimization"
 ],
 "ml_areas": [
  "world-model",
  "training-dynamics",
  "neural-ode"
 ],
 "paper": {
  "arxiv_id": "2608.22112",
  "arxiv_url": "https://arxiv.org/abs/2608.22112",
  "summary_what_math_gives_to_ml": "The paper provides two transferable constructions beyond an ordinary neural ODE: explicit Runge–Kutta composition trained over recursively generated trajectories, and a compositional symbolic network whose primitives are linear combinations and learnable products of powers. The important asset is not merely interpretability; multi-step composition directly trains the map that will be used at inference time, while sparsity can compress the resulting vector field into a small set of terms. A practical transfer is to use horizon curricula and rollout-consistency losses for learned dynamics modules, and to use the symbolic primitive network as a sparse replacement for an MLP when the target dynamics are low-dimensional or structured.",
  "title": "Symbolic Neural ODEs: Learning interpretable models from time-series data",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 5,
  "novelty": 5,
  "usefulness": 8
 },
 "solves": [
  "stability",
  "accuracy",
  "sample-efficiency"
 ],
 "title": "Progressive rollout-consistency training",
 "url": "https://synthcore.org/idea/44/progressive-rollout-consistency-training",
 "verification": {
  "peer_reviewed": false,
  "stage1_mechanism_check": {
   "worked": false,
   "confidence": 9,
   "verdict": "Built differentiable RK4 rollout training with L1 regularization, progressive horizons, gradient clipping, synthetic noisy nonlinear oscillator data, and long-horizon evaluation. The corrected numerical sanity check confirmed fourth-order RK4 scaling (error ratios 17.09 and 16.59 under timestep halving), but at approximately equalized training evaluations the one-step baseline was better at every horizon: 64-step MAE 0.04368 versus 0.07336 for progressive training. No trajectories diverged in either method, so the promised forecasting/stability win was not observed in this MVP.",
   "metrics": {
    "baseline": "MAE at horizons 8/16/32/64: 0.01121/0.01994/0.03112/0.04368; max norm 1.586; divergence rate 0.0",
    "idea": "MAE at horizons 8/16/32/64: 0.01759/0.03150/0.05122/0.07336; max norm 1.591; divergence rate 0.0; RK4 halving ratios 17.09 and 16.59"
   },
   "how_to_run": "python3 experiment.py",
   "files": [
    "experiment.py",
    "results.json"
   ],
   "limitations": "Only one seed and one stable 2D synthetic oscillator were tested; no pendulum, Van der Pol, Lorenz, multiple noise levels, hyperparameter tuning, wall-clock/FLOP profiling, sparse-model recovery, or statistical significance analysis was performed."
  },
  "status": "mechanism_failed",
  "status_label": "Mechanism failed",
  "updated_at": "2026-08-29T19:14:28",
  "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
   }
  }
 }
}