Эксперимент: Differentiable Physics-Equilibrium Projection (#1051)

{ "worked": true, "confidence": 9, "verdict": "Built a differentiable cubic-equilibrium projection with fixed-step Newton forward solving and an analytic implicit backward pass. The mechanism manifested: the least-squares iteration matched the predicted contraction factor |1-taua^2| and switched from contraction to divergence at the predicted boundary taua^2=2; implicit sensitivities matched finite differences, including 1/sigma_min(J) growth from 10 to about 999 as sigma_min(J) decreased from 0.1 to 0.001. Projection reduced the test residual to 3.93e-12 versus 0.109 for penalty training and 0.286 for direct prediction, while projected state MSE was 8.46e-16.", "metrics": { "baseline": "Direct predictor: mean test |F|=0.285812, MSE=0.009461. Penalty predictor: mean test |F|=0.109087, MSE=0.001355.", "idea": "Newton-projected penalty predictor: mean test |F|=3.929e-12, MSE=8.456e-16. Contraction ratios matched theory within about 6e-8 in the sweep; stability held for tau*a^2<2 and failed at 2.1. Implicit autograd derivative max absolute error was 0.0." }, "how_to_run": "/home/maxwelhelp/main/bin/python3 experiment.py", "files": [ "experiment.py", "implicit_layer.py", "results.json" ], "limitations": "This is a scalar cubic toy system, not a sparse AC 14-bus network, GNN, inequality restoration, or a wall-clock/FLOP comparison. The experiment does not test large-scale sparse linear solves, multiple equilibria, solver failures under realistic topology outages, or robustness to noisy/extrapolating inputs." }