Can a learned score see the margin of the arrow of time?

Abstract

An observer who sees a Hamiltonian ensemble through a Gaussian window of width σ(t) = σ₀e^{rt} finds that the coarse-grained entropy grows if and only if r exceeds a critical rate R of the state, and R stays below the largest strain rate κ of the flow by a margin proportional to σ. The same forward process, a flow followed by a widening blur, defines a phase-space diffusion model. We ask whether a score learned from samples, as in a diffusion model, can see this margin. Two thresholds must be told apart: a stochastic differential equation that uses the physical flow as its drift exists only for r ≥ κ, at every σ, while the coarse-grained entropy keeps growing down to r = R < κ. We estimate R in an integration-by-parts form in which each learned field appears once; we prove that errors orthogonal to the true fields leave this form exact while they bias the direct form, and find that for smooth random errors it is up to sixty times more accurate. Small networks trained by denoising score matching read the margin (κ − R)/σ to within 2.2% at 36 points of a pendulum and to within 1.4% at 24 points of two coupled pendulums, where they agree with the matrix formula for the resolution gap, including its finite-resolution correction, to within 2%. In the band between the two thresholds the stochastic equation with the physical drift departs from the true law while the entropy, measured directly with nearest-neighbour estimators, grows as predicted. The reverse equation fails exactly at r = κ, and the apparent knee slightly below κ follows from the shape of the missing diffusion. Code, trained networks and verification: https://github.com/neginyan/learned-score-arrow-of-time (doi:10.5281/zenodo.23407853).