Identifying knot types of polymer conformations by machine learning

We investigate the use of artificial neural networks (NNs) as an alternative tool to current analytical methods for recognizing knots in a given polymer conformation. The motivation is twofold. First, it is of interest to examine whether NNs are effective at learning the global and sequential properties that uniquely define a knot. Second, knot classification is an important and unsolved problem in mathematical and physical sciences, and NNs may provide insights into this problem. Motivated by these points, we generate millions of polymer conformations for five knot types: 0, $3_1, 4_1, 5_1$, and $5_2$, and we design various NN models for classification. Our best model achieves a five-class classification accuracy of above $99\%$ on a polymer of 100 monomers. We find that the sequential modeling ability of recurrent NNs is crucial for this result, as it outperforms feed-forward NNs and successfully generalizes to differently sized conformations as well. We present our methods and suggest that deep learning may be used in specific applications of knot detection where some error is permissible. Hopefully, with further development, NNs can offer an alternative computational method for knot identification and facilitate knot research in mathematical and physical sciences.

Figure 1

Two visual representations of the same circular polymer conformation in simulation. The task of this work is to recognize knots in such a circular polymer conformation. (a) Touching-bead model of a circular polymer in simulation. (b) The representation of a polymer conformation by rods.

Figure 2

Five knot types that are to be classified. The five diagrams at the top display the simple representations of these five knot types. The five diagrams at the bottom are examples of conformations generated by our Monte Carlo simulations. The knot types of these conformations are usually too complicated to be identified visually, and we train neural networks to recognize knots in these circular polymers.

Figure 3

Left: An example of polymer conformation with $L_{\mathrm{polymer}}=100$. Right: The inputs of this conformation for neural networks. $\mathrm{\Delta }x,\mathrm{\Delta }y,\mathrm{\Delta }z$ represent the changes in monomer positions in the $x, y$, and $z$ directions at every step, e.g., from the fifth to the sixth monomer.

Figure 4

The feed-forward architecture used in this work. 3D coordinates are flattened and sent through layers of $p,q,\cdots ,r$ fully connected neurons, abbreviated FC($p$), FC($q$), FC($r$), etc. The output layer contains sigmoid-activated neurons for $C$ classes, performing the softmax operation to output class probabilities for $y_c$. Please see Appendix pp2 for technical details on neural networks.

Figure 5

LSTM-based recurrent neural network used in our work. Input $x_i^{\left(t\right)}$ represents the $t\mathrm{th}$ step of input sequence $x_i$. The first layer of LSTM($p$) cells accepts three-dimensional inputs, and produces $p$-dimensional outputs, which are passed downstream. The LSTM($q$) layer is bidirectional, which scans the sequence forward and backward. Data flow from layer to layer, as well as between cells within layers, capturing sequentiality. The last layer contains sigmoid neurons, outputting $C$ class probabilities $y_c$, basing its predictions on the hidden state of the final recurrent layer's $T\mathrm{th}$ cell.

Figure 6

Feed-forward and RNN models trained with polymers of length $L_{\mathrm{polymer}}=100$ were used to predict five knot types on 1 million unseen test data. Predicted labels of the five knot types were plotted against true labels in a confusion matrix. The five knot type labels are knot 0, $3_1, 4_1, 5_1$, and $5_2$. The cells in the confusion matrix contain both the percentage of accurate predictions and the absolute number of data instances in parentheses. The colorbar represents percentage values from 0 to 1 with a blue colormap. (a) The feed-forward network is able to predict with accuracy between $67.6\%$ and $75.3\%$; (b) the RNN model is able to classify the polymers to their correct knot types consistently at above $99\%$.

Figure 7

The LSTM-based RNN model from Fig. 6 generalizing to predict knot types on polymers of variable length. The model was trained using polymer length $L_{\mathrm{polymer}}=100$ and was asked to predict knot type labels on 1 million $L_{\mathrm{polymer}}=60$ polymers and 1 million $L_{\mathrm{polymer}}=80$ polymers. Predicted labels of the five knot types were plotted against true labels in the confusion matrix. The five knot type labels are knot 0, $3_1, 4_1, 5_1$, and $5_2$. The cells in the confusion matrix contain both the percentage of accurate predictions and the absolute number of data instances in parentheses. The colorbar represents percentage values from 0 to 1 with a yellow-green colormap. Left is the confusion matrix for $L_{\mathrm{polymer}}=60$ and right is for $L_{\mathrm{polymer}}=80$, and both generalizations yield good accuracy.

Figure 8

Comparison of a real classification and a control experiment. Knot0-Knot31 refers to the data set of 1 million conformations of unknot and 1 million conformations of knot $3_1$ (a real classification). Knot31-Knot31 refers to the data of 2 million conformations of knot $3_1$ with two dummy labels (a control experiment). An epoch is an iteration over the entire training and validation data. Each epoch trains on 1 600 000 samples and validates on 400 000 samples.

Figure 9

Top: Evolution of the crossing number of the knot type, $N_{\mathrm{cross}}$, during the Monte Carlo simulation. Here, $N_{\mathrm{cross}}$ represents the minimum crossing number for a knot type, e.g., 5 for the $5_2$ knot. Middle: The autocorrelation of $N_{\mathrm{cross}}$ as a function of the simulation time. The correlation time is approximately $5.4\times {10}^3$ steps. Bottom: The distribution of knot types in our simulation of the polymer $L=100$ and $L_p=4a$ confined in a sphere with a diameter of $D=11a$.

Figure 10

The neuron calculates a weighted sum of the inputs, and passes the result through a nonlinear activation function $a(w^{T}x+b)$, to scale the output in the $[\mathbf{0},\mathbf{1}]$ range. Many neurons are connected together in a neural network.

Figure 11

The confusion matrix of applying the RNN model trained from the conformations of $L_p=4a$ to the conformations with $L_p=2a$.