Эксперимент: Padé-Hermite Neural ODE Integrator (#534)

{ "worked": true, "confidence": 8, "verdict": "Built an s=2 Padé-Hermite anchored two-derivative integrator with coefficients generated from the paper's polynomial-basis construction, plus numerical verification and stiff toy comparisons. The mechanism manifested: moment residual was 1.1e-16, observed order was 3.94/3.96/3.78 versus predicted order 4, the left-half-plane grid maximum was |R|=1.0 versus A-stability, and the negative-axis decay slope was -1.96 versus the predicted -2. Node-independent amplification matched to about 1e-15, while nonlinear fixed-point convergence failed at larger stiff steps, so the stability claim is supported but practical nonlinear robustness is not.", "metrics": { "baseline": "Explicit Euler stability boundary observed at z=-2: |1+z| was 0.9 at -1.9, 1.0 at -2.0, and 1.1 at -2.1. RK4 on the nonlinear lambda=80 test failed at h=0.05, had error 6.77e-8 at h=0.025, and error 2.24e-9 at h=0.0125.", "idea": "s=2 Padé-Hermite: observed global orders 3.94, 3.96, 3.78 versus predicted 4; |R(-100)|=5.42e-4 and |R(-1000)| approximately 5.42e-6, giving observed log-log slope -1.96 versus predicted -2; amplification was identical across internal nodes [0.5, 0.25, 0.75] within roughly 1e-15. On the nonlinear stiff test, h=0.05 and 0.025 failed the simple damped fixed-point solve, while h=0.0125 converged with error 4.59e-10 but required 1194 total iterations." }, "how_to_run": "/home/maxwelhelp/main/bin/python3 experiment.py", "files": [ "experiment.py", "results.json" ], "limitations": "Only a small CPU NumPy toy experiment was tested; no neural ODE, JVP implementation, CIFAR-10 probability-flow model, GPU timing, adaptive step controller, Newton/Krylov solve, FLOP-normalized comparison, or sample-quality metric was evaluated. The A-stability check was a finite complex grid, and nonlinear failures reflect the basic fixed-point solver rather than the exact linear stability function." }