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