Optimal preparation and reachable-state constraints in the Mpemba effect

What the math gives to ML

The paper offers a transferable optimal-control mechanism: the preparation protocol, rather than only the prepared state, determines the strongest possible anomalous relaxation, and Pontryagin's maximum principle predicts a one-bang control. Its key engineering asset is a reachable-state bound: a stochastic thermostat cannot create arbitrary non-Gaussianity, so the achievable relaxation-rate improvement is limited by the dynamics and control amplitude. A neural-network analogue is to control gradient noise or learning-rate modulation during a short preparation phase, targeting a beneficial gradient-distribution kurtosis while respecting empirically estimated reachability bounds, and then test whether the optimal schedule is boundary-valued with a single switch rather than smoothly annealed.