Эксперимент: Regularity-Matched Random Fourier Layer (#1327)

{ "worked": true, "confidence": 8, "verdict": "Built a fixed-seed Regularity-Matched RFF MVP with Gaussian, uniform-window, and Gevrey-matched frequency sampling, ridge linear fitting, empirical whitening, and distribution/conditioning sanity checks. The matched layer clearly improved low-feature approximation: RMSE was 0.0501 vs 0.402 for Gaussian at N=64, and 0.00234 vs 0.01355 at N=128 after whitening. The effect is not universal—Gaussian caught up or exceeded it at N>=256—and whitening reduced error but did not fully eliminate severe Gram ill-conditioning, so this is a promising low-N signal rather than a broad win.", "metrics": { "baseline": "Gaussian RFF RMSE: N=64 0.4020, N=128 0.01355, N=256 0.00010, N=512 approximately 0; raw Gram condition numbers approximately 3.9e2, 3.6e6, 8.2e15, 1.2e16.", "idea": "Gevrey-matched RFF RMSE: N=64 0.05006 raw / 0.04531 whitened, N=128 0.00954 raw / 0.00234 whitened, N=256 0.00353 raw / 0.00052 whitened, N=512 0.00034 raw / 0.00002 whitened; raw Gram condition numbers approximately 5e12, 2.5e16, 4.6e16, 9.5e16. Empirical radii had Gaussian median 3.53 versus matched median 1.12, consistent with concentrating frequencies near the smooth target’s spectral region." }, "how_to_run": "/home/maxwelhelp/main/bin/python3 experiment.py > run_output.txt", "files": [ "experiment.py", "results.json", "run_output.txt" ], "limitations": "Only a single analytic 2D synthetic target was tested, with three random repetitions per configuration; no PDE, operator-learning, grid-based 64x64 experiment, FLOP-normalized timing, ultra-analytic distribution, or larger-dimensional target was evaluated. The current ridge-style whitening improves prediction but does not guarantee a well-conditioned transformed Gram matrix for nearly redundant features." }