Interpretable machine learning model for the deformation of multiwalled carbon nanotubes

We present an interpretable machine learning model to predict accurately the complex rippling deformations of multiwalled carbon nanotubes made of millions of atoms. Atomistic-physics-based models are accurate but computationally prohibitive for such large systems. To overcome this bottleneck, we have developed a machine learning model that comprises a novel dimensionality reduction technique and a deep neural network-based learning in the reduced dimension. The proposed nonlinear dimensionality reduction technique extends the functional principal component analysis to satisfy the constraint of deformation. Its novelty lies in designing a function space that satisfies the constraint exactly, which is crucial for efficient dimensionality reduction. Owing to the dimensionality reduction and several other strategies adopted in the present paper, learning through deep neural networks is remarkably accurate. The proposed model accurately matches an atomistic-physics-based model whereas being orders of magnitude faster. It extracts universally dominant patterns of deformation in an unsupervised manner. These patterns are comprehensible and explain how the model predicts yielding interpretability. The proposed model can form a basis for an exploration of machine learning toward the mechanics of one- and two-dimensional materials.

Figure 1

Schematic of the present framework involving the data generation via AC simulation and the proposed DML model. The DML model includes: c FPCA and DNN. Firm arrows show data generation and training, and dashed arrows show prediction via the proposed model.

Figure 2

Illustration of the kinematics showing the configuration spaces, deformation maps, and coordinate axes.

Figure 3

Cross sections of MWCNT obtained using two approaches of the DML model ($\begin{array}{c}- - -\end{array}$) that uses FPCA: (a) FPCA coupled with DNN and (c) FPCA coupled with the constrained DNN. These two approaches are compared against the AC model ($\begin{array}{c}———\end{array}$). (b) and (d) are the close-up views of the blue boxes corresponding to (a) and (c). (b) Shows a discontinuity in the DML model and (d) shows increased error in predictions everywhere.

Figure 4

Cumulative percent variance captured by principal components for MWCNTs under torsion (left) and bending (right).

Figure 5

Correlation plots for the test set CoFPCS of (a) in-plane deformation in torsion, (b) in-plane and (c) out-of-plane deformation in bending. $R=0.9943\left(\mathrm{a}\right),0.9931\left(\mathrm{b}\right),0.9991\left(\mathrm{c}\right)$.

Figure 6

(a) Twisted 40-walled CNT obtained via AC (top) and DML (bottom) models. (b) Radial deformation color map (red: high; blue: low). Alternate walls of cross sections obtained via AC ($\begin{array}{c}———\end{array}$) and DML ($\begin{array}{c}- - -\end{array}$) models, for 10, 20, 30, and 40-walled CNTs.

Figure 7

Bent 35-walled CNT obtained via AC (a) and (c) and DML (b) and (d) models. (c) and (d) show color maps of radial deformations corresponding to (a) and (b). (e)–(g) Alternate walls of cross sections obtained via AC ($\begin{array}{c}———\end{array}$) and DML ($\begin{array}{c}- - -\end{array}$) models.

Figure 8

Comparison of AC (top) and DML (bottom) models for a 32-walled CNT (system which is not a part of the training data) under torsion (a) and bending (b).

Figure 9

Energy comparison for (15-, 25-, 32-, 40-walled) MWCNTs under torsion (a) and bending (b) via AC ($\begin{array}{c}○\end{array}$) and DML models ($+$). A 32-walled CNT ($\begin{array}{c}+\end{array}$) is an unknown system. The lines ($\begin{array}{c}———\end{array}$) and ($\begin{array}{c}———\end{array}$) are drawn to highlight pre- and postbuckling regimes.

Figure 10

Functional principal components of MWCNTs under torsion (a) and (b) and bending (c) and (d).