A machine-learned dynamical phase diagram of the one-dimensional nonlinear Schr\"odinger equation with quasiperiodic disorder and a static field

Abstract

We map the transport regimes of a wave packet evolving under the one-dimensional discrete nonlinear Schr\"odinger (Gross-Pitaevskii) equation with a quasiperiodic Aubry-Andr\'e potential and a static (Stark) field, as a function of self-interaction $g$, field $F$, and quasiperiodic strength $\lambda$, for two initial states (delta and Gaussian). Instead of one expensive long-time simulation per grid point, we run 3600 short simulations, reduce each to eight dynamical features, and cluster parameter space with a Gaussian mixture model. Clustering finds the boundaries between regimes unsupervised; each cluster's name is then assigned by a hand-calibrated rule, and an ablation shows clustering measurably smooths those boundaries rather than merely relabeling them. The five resulting regimes (ballistic, localized, oscillatory-localized, subdiffusive, and self-trapped) are validated against 100 long runs on a lattice eight times larger than the short sweep, so the validation window is longer than the training window even for the fastest-spreading regime. The short-time clusters predict the long-time asymptotic spreading exponent ($\alpha_\infty = 2.01\pm0.07$ ballistic, $0.28\pm0.16$ subdiffusive, consistent with the weak-chaos prediction $\alpha=1/3$) and, independently, a long-time retention $\Pi_0=0.72\pm0.32$ for the self-trapped cluster, whose exponent alone is not diagnostic. The initial-state contrast is striking: the delta state develops a broad self-trapping wedge (onset at $g=3.907\pm0.017$ at $\lambda=0$) that invades the ballistic and localized regions as $g$ grows, while the Gaussian state shows no self-trapping, instead opening a subdiffusive corridor along the Aubry-Andr\'e critical line that widens with $g$. The scheme yields a full $15\times15$ phase map in minutes per slice, where direct asymptotic simulation ($t\gtrsim10^6$ per point) is prohibitive.