Эксперимент: Energy-trained monotone coordinate warp (#518)

{ "worked": true, "confidence": 8, "verdict": "Built a readable MVP for positive-density monotone radial warps, with normalized cumulative integration, exponent and interpolation-convergence sweeps, and a small learned-warp neural regression comparison. The mechanism manifested: fitted r(s) exponents matched q, transformed singular exponents matched qλ within about 0.02%, and interpolation error slopes matched the predicted -qλ values within about 0.015. The learned warp slightly improved held-out RMSE over the fixed s² warp (0.00430 vs 0.00465), but required substantially more time and this is not evidence of a PDE-level win.", "metrics": { "baseline": "Identity warp RMSE 0.02543 in 500 steps, 1.45 s; fixed r=s² warp RMSE 0.00465, 1.49 s.", "idea": "Learned positive-density warp RMSE 0.00430 in 500 steps, 7.10 s; exponent law observed qλ: qλ=0.25, 0.50, 0.75 gave convergence slopes -0.255, -0.510, -0.765 versus predictions -0.25, -0.50, -0.75; exponent-fit errors were below roughly 0.02%." }, "how_to_run": "/home/maxwelhelp/main/bin/python3 experiment.py", "files": [ "experiment.py", "results.json", "run_output.txt" ], "limitations": "The mini-experiment is a 1D singular-function regression rather than the proposed 2D L-shaped Poisson Deep Ritz/PINN problem. It does not test PDE energy, polar Jacobian derivative transforms, boundary conditions, changing coefficients, GPU execution, or an outer unrolled optimization with a held-out residual; finite-difference derivative relative errors near vanishing densities are ill-conditioned." }