Detecting depinning and nonequilibrium transitions with unsupervised machine learning

Using numerical simulations of a model disk system, we demonstrate that a machine learning generated order-parameter-like measure can detect depinning transitions and different dynamic flow phases in systems driven far from equilibrium. We specifically consider monodisperse passive disks with short range interactions undergoing a depinning phase transition when driven over quenched disorder. The machine learning derived order-parameter-like measure identifies the depinning transition as well as different dynamical regimes, such as the transition from a flowing liquid to a phase separated liquid-solid state that is not readily distinguished with traditional measures such as velocity-force curves or Voronoi tessellation. The order-parameter-like measure also shows markedly distinct behavior in the limit of high density where jamming effects occur. Our results should be general to the broad class of particle-based systems that exhibit depinning transitions and nonequilibrium phase transitions.

Figure 1

(a) The average disk velocity $\langle V_x\rangle$ vs driving force $F_D/F_p$ in samples with total disk density $\varphi$ of $\varphi =0.85$ (down triangles), 0.71 (pentagons), 0.61 (right triangles), 0.55 (stars), 0.43 (squares), 0.30 (up triangles), and 0.25 (circles). (b) The corresponding $P_6$ vs $F_D/F_p$.

Figure 2

Images obtained at a disk density of $\varphi =0.30$ for the system in Fig. 1 at $F_D=0.25$ (pinned), (b) $F_D=0.95$ (phase separated), (c) $F_D=1.5$ (smectic flow), and (d) $F_D=2.5$ (smectic flow).

Figure 3

(a) The machine learning derived order-parameter-like measure $P_1$ vs $F_D/F_p$ for the system in Fig. 1 at disk density $\varphi =0.30$ with 50 realizations. Three phases are clearly apparent: the pinned state at $0<F_D/F_p<0.65$, the phase separated state at $0.65\le F_D/F_p<1.3$, and the smectic or laned state at $F_D\ge 1.3$. (b) $d\langle V_x\rangle /d{F}_D$ vs $F_D/F_p$ for the same system.

Figure 4

The scaled eigenvalues (relative scores) ${\lambda }_N$ of the PCA algorithm vs the relative ranking $N$ for disk densities of $\varphi =0.85$ (down triangles), 0.71 (pentagons), 0.61 (right triangles), 0.55 (stars), 0.43 (squares), 0.30 (up triangles), and 0.25 (circles). $N=1$ corresponds to the first principal component.

Figure 5

$P_1$ vs $F_D/F_p$ for the system in Fig. 1 at intermediate disk densities using 50 realizations. (a) At $\varphi =0.43$, there are multiple peaks. (b) At $\varphi =0.61$, the peak structure is more compressed. (c) $d\langle V_x\rangle /F_D$ vs $F_D/F_p$ for the same systems at $\varphi =0.43$ (blue squares) and $\varphi =0.61$ (orange triangles).

Figure 6

Images obtained at a disk density of $\varphi =0.43$ for the system in Fig. 5 at (a) $F_D/F_p=0.05$ (disordered pinned state), (b) $F_D/F_p=0.55$ (clustering clogged state), (c) $F_D/F_p=0.75$ (moving liquid), (d) $F_D/F_p=0.95$ (phase separated state with amorphous order), (e) $F_D/F_p=1.5$ (phase separated state with crystalline order), and (f) $F_D/F_p=2.5$ (moving smectic).

Figure 7

Images obtained at a disk density of $\varphi =0.61$ for the system in Fig. 5 at (a) $F_D/F_p=0.25$ (pinned clogged state), (b) $F_D/F_p=0.95$, (phase separated with crystalline order), (c) $F_D/F_p=1.5$ (moving state), and (d) $F_D/F_p=2.5$ (moving state).

Figure 8

$P_1$ vs $F_D/F_p$ for the system in Fig. 1 at $\varphi =0.85$ using 50 realizations, where the disks exhibit jamming behavior and elastic depinning.

Figure 9

Images obtained at a disk density of $\varphi =0.85$ for the system in Fig. 8, where the system forms a jammed solid with increasing triangular ordering at higher drives. (a) $F_D/F_p=0.25$. (b) $F_D/F_p=2.5$.

Figure 10

First vector $\left[q_1\right]$ from the total transformation matrix $\vec{Q}$ obtained from 50 realizations versus $k$, which is related to a neighbor distance, for the system in Fig. 1 at disk densities $\varphi =$ (a) 0.25, (b) 0.43, (c) 0.61, and (d) 0.85.

Figure 11

Finite size analysis of density $\varphi =0.30$ for system of varying sizes. $P_1$ vs $F_D/F_p$ for $S_x=20$ (light blue circles), 40 (green squares), 60 (dark blue triangles), and 80 (red diamonds).