# Эксперимент: Harmonic-coordinate neural PDE ansatz (#372)

{
  "worked": true,
  "confidence": 8,
  "verdict": "Built a PyTorch harmonic-coordinate PDE ansatz with autodiff Laplacian and complex null-gradient residuals, plus a direct MLP baseline. The toy sweep confirmed the mechanism: the canonical q=x+iy identity had zero residual, the perturbed coordinate harmonic defect was exactly 2ε, and the null-gradient defect scaled as ε with observed exponent 1.000 and coefficient 2.0645, matching 4E|x|=2.0645. The ansatz was not an optimization win in this tiny test: baseline test MSE/Laplacian residual were 6.72e-5/0.0438 versus 2.56e-4/0.0875 for the ansatz.",
  "metrics": {
    "baseline": "test MSE 6.7221e-05; mean absolute Laplacian 0.04380",
    "idea": "test MSE 2.5562e-04; mean absolute Laplacian 0.08745; coordinate-constraint RMS 0.09148; defect scaling exponent 1.000"
  },
  "how_to_run": "cd /home/maxwelhelp/all/math2nn/experiments/exp892_harmonic-coordinate-neural-pde-ansatz && /home/maxwelhelp/main/bin/python3 experiment.py",
  "files": [
    "experiment.py",
    "results.json",
    "run_output.txt"
  ],
  "limitations": "Only a small 2D manufactured harmonic problem was tested, with one seed, 500 optimization steps, and no Fourier-feature or Poisson benchmark. The mechanism check verifies coordinate defect scaling but does not test broad PDE families, freezing or pretraining q, equal-FLOP comparisons, or sparse-collocation generalization."
}