# Эксперимент: Sobolev-Calibrated Frozen Sigmoid Features (#992)

{
  "worked": false,
  "confidence": 8,
  "verdict": "Built frozen tanh Sobolev features with analytic first derivatives, Sobolev-weighted linear least squares, random/grid parameter sampling, and numerical evaluation. The derivative identity was verified to about 6e-9 error, and 2D errors decreased with feature count, but observed slopes (-0.714 for values and -0.392 for gradients) did not adequately confirm the predicted d=2,k=2 rates (-1 and -0.5); random-versus-grid behavior was also non-monotonic rather than a clean logarithmic penalty. The mechanism shows a promising approximation signal, but the required quantitative predictions were not sufficiently confirmed, so this is not counted as a win.",
  "metrics": {
    "baseline": "The earlier 1D trainable tanh baseline at M=64 had value RMSE 0.1195, gradient RMSE 0.8845, Hessian RMSE 10.25, and about 17.38 seconds per seed; this comparison is not treated as decisive because the 1D frozen fit became nearly exactly representable.",
    "idea": "In the controlled 2D sweep, random frozen Sobolev features improved from value/gradient RMSE 0.757/3.689 at M=8 to 0.07396/0.9648 at M=192; fitted slopes were -0.714/-0.392 versus predicted -1/-0.5."
  },
  "how_to_run": "/home/maxwelhelp/main/bin/python3 sobolev_2d_check.py",
  "files": [
    "sobolev_features.py",
    "sobolev_2d_check.py",
    "results.json",
    "results_2d.json"
  ],
  "limitations": "No PDE/PINN experiment, CUDA implementation, higher-dimensional scaling study, or robust trainable 2D baseline was tested. The theorem provides upper bounds rather than guaranteed empirical slopes, and the grid construction used a truncated tensor parameter grid that is not an optimized deterministic dictionary."
}