Recognition of polymer configurations by unsupervised learning

Unsupervised learning as an important branch of machine learning is commonly adopted to discover patterns, with the purpose of conducting data clustering without being labeled in advance. In this study, we elucidate the striking ability of unsupervised learning techniques in exploring the phase transitions of polymer configurations. In order to extract the low-dimensional representation of polymer configurations, principal component analysis and diffusion map are applied to distinguish the coiled state and collapsed states and further detect the delicate distinction among collapsed states, respectively. These dimensionality reduction techniques not only identify the distinct states in the feature space, but also offer significant insights to understand the relation between salient features and order parameters in physics. In addition, a hybrid neural network scheme combining the supervised learning and unsupervised learning is utilized to precisely detect the critical point of phase transition between polymer configurations. Our study demonstrates a promising strategy based on the unsupervised learning, particularly in the exploration of phase transition in polymeric systems.

Figure 1

The characteristic configurations of a polymer chain in (a) coil, (b) globule, (c) anti-Mackay, and (d) Mackay states as the energy is lowered. The detailed monomer packing modes for liquid-like globule state and crystal-like anti-Mackay and Mackay states are shown in (e), (f), and (g), respectively. For the globule state (e), monomer positions are disordered. Anti-Mackay and Mackay states differ from each other by the monomer stacking symmetries, demonstrated by the red (dark gray) and yellow (light gray) spheres in panels (f) and (g).

Figure 2

The top eigenvalues ${\lambda }_l$ obtained from PCA for coil and globule states in the reduced energy range $e\in [-3,1]$.

Figure 3

Two-dimensional projections of configurational data into the top three eigenvectors (a) $W_1\sim W_2$, (b) $W_1\sim W_3$, and (c) $W_2\sim W_3$ from PCA in the reduced energy range $e\in [-3,1]$.

Figure 4

The behaviors of the nonlinear transformation of the top $K$ (PCA) eigenvectors are described according to ${\sum }_{l=1}^K{W}_l^2$. (a) The mapping between ${\sum }_{l=1}^{K=3}{W}_l^2$ and $R_g^2/{\sigma }^2$. The insert plot is for $K=9$. (b) The Pearson correlation coefficient $\rho ({\sum }_{l=1}^K{W}_l^2,R_g^2/{\sigma }^2)$ as a function of $K$. $R_g$ is the radius of gyration for polymers.

Figure 5

The two-dimensional projection of the embedding configurations constituting of globule, anti-Mackay, and Mackay into the top two nontrivial leading eigenvectors $({\lambda }_2{\psi }_2\sim {\lambda }_3{\psi }_3)$ obtained from diffusion map. Data are colored by (a) configuration density $p\left(x\right)$, (b) energy $e=E/N\epsilon$, (c) the local order parameter $Q_6^{\mathrm{core}}$, and (d) the global order parameter $Q_6$, respectively. $p\left(x\right)$ is the normalized histogram to the configuration density at $x$. The details to calculate $Q_6^{\mathrm{core}}$ and $Q_6$ can be found in Appendix pp3.

Figure 6

The structural transition determined by the W-shape approach in terms of input configurations pre-processed by PCA with truncation of dimensionality $l$ as labeled in plots. The critical energy $e_c$ positioned by the dashed lines for (a) coil-to-globule (b) globule-to-anti-Mackay, and (c) anti-Mackay-to-Mackay transitions reads $-1.61, -4.43$, and $-4.74$, respectively.