Machine learning topological defects of confined liquid crystals in two dimensions

Supervised machine learning can be used to classify images with spatially correlated physical features. We demonstrate the concept by using the coordinate files generated from an off-lattice computer simulation of rodlike molecules confined in a square box as an example. Because of the geometric frustrations at high number density, the nematic director field develops an inhomogeneous pattern containing various topological defects as the main physical feature. We describe two machine-learning procedures that can be used to effectively capture the correlation between the defect positions and the nematic directors around them and hence classify the topological defects. First is a feedforward neural network, which requires the aid of presorting the off-lattice simulation data in a coarse-grained fashion. Second is a recurrent neural network, which needs no such sorting and can be directly used for finding spatial correlations. The issues of when to presort a simulation data file and how the network structures affect such a decision are addressed.

Figure 1

(a–d) Illustration of a two-dimensional handwritten image, Ising model, nematic state, and coordinates for a rodlike molecule, as well as (e–h) example configurations of different defect states generated from Monte Carlo simulations for a confined rodlike nematic fluid. The parameters used to generate these example configurations are $[N,a/L]=[784,6.32]$. The defect state in (e) has a diagonal (D) pattern, and the nematic textures in (f–h) resemble the tilted letter T, letter U, and letter X. The yellow and blue circles mark the defect locations of $-1$ and $-1/2$ winding numbers, respectively.

Figure 2

Schematics of (a) an FNN with two hidden layers and the sequence $(x_l,y_l,{\theta }_l)$ as input, where $l=1,2,...N$, and (b) an RNN (unrolled) with the sequence $x_l$ ($l=1,2,...N$) as the input to the first LSTM block, $y_l$ ($l=1,2,...N$) to the second LSTM block, and ${\theta }_l$ ($l=1,2,...N$) to the third LSTM block. LSTM blocks also have LSTM-LSTM connections. The final output of the two networks are ${\nu }_1,{\nu }_2,...,{\nu }_n$, where $n$ is adjusted according to the type of features to be identified.

Figure 3

Cost function $S$ and accuracy $A$, defined on the training and test datasets respectively, monitored on an FNN [(a) and (b)] and RNN [(c) and (d)] as functions of epoch step. Symbols in the plots represent the averaged $S$ and $A$ produced from 20 repeated training runs, from which errorbars are also estimated. Circles, squares, triangles, and diamonds in (a) and (b) correspond to the degrees of coarse-graining in the presorting procedure used: $m=1$ (unsorted raw data), 2, 4, and 8, respectively. For coarse-graining, the original simulation box in Figs. 1–1 is divided into $m\times m$ cells [see Sec. 3]. The circles in (c) and (d) also represent the same averaged $S$ and $A$, where an RNN was used with raw, unsorted data as the input (i.e., $m=1$).

Figure 4

Identification of the I-N transition point. An FNN is trained by using training files at low and high densities ($\rho =4$ and 16 for the I and N states, respectively). Then, the trained FNN is used to calculate the average outputs $\overline{{\nu }_1}$ and $\overline{{\nu }_2}$ at each given $\rho$. These characteristic measures are to indicate if the system is in an I or N state, plotted by red and white squares, respectively. Error bars of these data points are smaller than the symbol. The crossing point is defined as the I-N transition point. In the background, represented by circles (to the right scale), an independent estimate of the variance of the orientational order parameter ${\sigma }^2$ (normalized) shows a peak at the transition point. The filled and unfilled circles represent the statistics from the rods in the entire box and half-sized center region, respectively. The three plots, (a), (b), and (c), are produced with system size $N={20}^2, {24}^2$, and ${28}^2$, respectively.

Figure 5

Demonstration of the importance of pixel ordering for MNIST recognition when FNN is used (circles) and the automatic pixel-position and feature correlation for MNIST recognition when RNN is used (squares). When the pixel ordering [green arrows in Fig. 1] is scrambled, an FNN cannot be trained adequately to recognize the MNIST images, shown by the rising cost and low accuracy on test data, even when the position-feature data are used together in the input to Fig. 2. However, when the same data are used with an RNN [Fig. 2] the network can successfully identify the 10 different digits, as shown by the minimal cost entropy and near-perfect test set accuracy.