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Abstract The ground state electron density — obtainable using Kohn-Sham Density Functional Theory (KS-DFT) simulations — contains a wealth of material information, making its prediction via machine learning (ML) models attractive. However, the computational expense of KS-DFT scales cubically with system size which tends to stymie training data generation, making it difficult to develop quantifiably accurate ML models that are applicable across many scales and system configurations. Here, we address this fundamental challenge by employing transfer learning to leverage the multi-scale nature of the training data, while comprehensively sampling system configurations using thermalization. Our ML models are less reliant on heuristics, and being based on Bayesian neural networks, enable uncertainty quantification. We show that our models incur significantly lower data generation costs while allowing confident — and when verifiable, accurate — predictions for a wide variety of bulk systems well beyond training, including systems with defects, different alloy compositions, and at multi-million-atom scales. Moreover, such predictions can be carried out using only modest computational resources. 

A novel machine learning model is presented in this work to obtain the complex high-dimensional deformation of Multi-Walled Carbon Nanotubes (MWCNTs) containing millions of atoms. To obtain the deformation of these high dimensional systems, existing models like Atomistic, Continuum or Atomistic-Continuum models are very accurate and reliable but are computationally prohibitive for these large systems. This high computational requirement slows down the exploration of physics of these materials. To alleviate this problem, we developed a machine learning model that contains a) a novel dimensionality reduction technique which is combined with b) deep neural network based learning in the reduced dimension. The proposed non-linear dimensionality reduction technique serves as an extension of functional principal component analysis. This extension ensures that the geometric constraints of deformation are satisfied exactly and hence we termed this extension as constrained functional principal component analysis. The novelty of this technique is its ability to design a function space where all the functions satisfy the constraints exactly, not approximately. The efficient dimensionality reduction along with the exact satisfaction of the constraint bolster the deep neural network to achieve remarkable accuracy. The proposed model predicts the deformation of MWCNTs very accurately when compared with the deformation obtained through atomistic-physics-based model. To simulate the complex high-dimensional deformation the atomistic-physics-based models takes weeks high performance computing facility, whereas the proposed machine learning model can predict the deformation in seconds. This technique also extracts the universally dominant pattern of deformation in an unsupervised manner. These patterns are comprehensible to us and provides us a better explanation on the working of the model. The comprehensibility of the dominant modes of deformation yields the interpretability of the model.