# Эксперимент: Feasible High-Order Neural ODE Solver (#1218)

{
  "worked": false,
  "confidence": 8,
  "verdict": "Built a two-stage Gauss-Legendre implicit solver and the paper-style box-constrained nonlinear least-squares variant. The unconstrained method numerically achieved fourth-order convergence, with observed orders 4.016, 4.004, 4.001, and 4.000, while the constrained method guaranteed zero stage and final-state violations. However, on the stiff scalar benchmark it often had larger residuals and worse error at coarse step sizes, with many inner iterations; clipping matched unconstrained final-state accuracy in this test. Feasibility was demonstrated, but the claimed accuracy or cost win was not.",
  "metrics": {
    "baseline": "Gauss implicit RK achieved fourth-order convergence; for lambda=40 and n=2, final error was 0.3013 with maximum stage violation 0.1077; for n=10, final error was 7.25e-12.",
    "idea": "The constrained Gauss solve had zero constraint violation; for lambda=40 and n=2, final error was 0.00224 with mean residual 0.2501 and 16 inner iterations; for n=10, final error was zero with mean residual 0.00547 and 45 inner iterations. For lambda=10 and n=2, constrained error was 0.0632 versus baseline error 0.0109."
  },
  "how_to_run": "/home/maxwelhelp/main/bin/python3 experiment.py && /home/maxwelhelp/main/bin/python3 sweep.py",
  "files": [
    "experiment.py",
    "sweep.py",
    "results.json",
    "sweep_results.json"
  ],
  "limitations": "Only a one-dimensional synthetic stiff system was tested; no neural vector field, MNIST or classification task, Radau method, multidimensional boxes, GPU timing, FLOP-matched training, or comparison against a projected Newton/Gauss-Newton solver was evaluated. The constrained optimizer uses bounded L-BFGS-B and can terminate at nonzero residual near active constraints."
}