Alexander V. Shapeev

V. Ehrlacher, C. Ortner, A. V. Shapeev

Numerical simulations of crystal defects are necessarily restricted to finite computational domains, supplying artificial boundary conditions that emulate the effect of embedding the defect in an effectively infinite crystalline environment. This work develops a rigorous framework within which the accuracy of different types of boundary conditions can be precisely assessed. We formulate the equilibration of crystal defects as variational problems in a discrete energy space and establish qualitatively sharp regularity estimates for minimisers. Using this foundation we then present rigorous error estimates for (i) a truncation method (Dirichlet boundary conditions), (ii) periodic boundary conditions, (iii) boundary conditions from linear elasticity, and (iv) boundary conditions from nonlinear elasticity. Numerical results confirm the sharpness of the analysis.

C. Ortner, A. V. Shapeev

We introduce a general class of (quasi-)interpolants of functions defined on a Bravais lattice, and establish several technical results for these interpolants that are crucial ingredients in the analysis of atomistic models and atomistic/continuum multi-scale methods.

Matthew Dobson, Christoph Ortner, Alexander V. Shapeev

We show under general conditions that the linearized force-based quasicontinuum (QCF) operator has a positive spectrum, which is identical to the spectrum of the quasinonlocal quasicontinuum (QNL) operator in the case of second-neighbour interactions. Moreover, we establish a bound on the condition number of a matrix of eigenvectors that is uniform in the number of atoms and the size of the atomistic region. These results establish the validity of and improve upon recent conjectures ([arXiv:0907.3861, Conjecture 2] and [arXiv:0910.2013, Conjecture 8]) which were based on numerical experiments. As immediate consequences of our results we obtain rigorous estimates for convergence rates of (preconditioned) GMRES algorithms, as well as a new stability estimate for the QCF method.

Christoph Ortner, Alexander V. Shapeev

We present a comprehensive a priori error analysis of a practical energy based atomistic/continuum coupling method (Shapeev, arXiv:1010.0512) in two dimensions, for finite-range pair-potential interactions, in the presence of vacancy defects. The majority of the work is devoted to the analysis of consistency and stability of the method. These yield a priori error estimates in the H1-norm and the energy, which depend on the mesh size and the "smoothness" of the atomistic solution in the continuum region. Based on these error estimates, we present heuristics for an optimal choice of the atomistic region and the finite element mesh, which yields convergence rates in terms of the number of degrees of freedom. The analytical predictions are supported by extensive numerical tests.

Olga Klimanova, Timofei Miryashkin, Alexander Shapeev

We present an algorithm for computing melting points by autonomously learning from coexistence simulations in the NPT ensemble. Given the interatomic interaction model, the method makes decisions regarding the number of atoms and temperature at which to conduct simulations, and based on the collected data predicts the melting point along with the uncertainty, which can be systematically improved with more data. We demonstrate how incorporating physical models of the solid-liquid coexistence evolution enhances the algorithm's accuracy and enables optimal decision-making to effectively reduce predictive uncertainty. To validate our approach, we compare the results of 20 melting point calculations from the literature to the results of our calculations, all conducted with same interatomic potentials. Remarkably, we observe significant deviations in about one-third of the cases, underscoring the need for accurate and reliable algorithms for materials property calculations.

Assyr Abdulle, Ping Lin, Alexander V. Shapeev

Among the efficient numerical methods based on atomistic models, the quasicontinuum (QC) method, introduced by Tadmor, Ortiz, and Phillips (1996), has attracted growing interest in recent years. Originally, the QC method was developed for materials with simple crystalline lattice (simple crystals) and later was extended to complex lattice (Tadmor et al, 1999). In the present paper we formulate the QC method for complex lattices in a homogenization framework and perform analysis of such a method in a 1D setting. We also present numerical examples showing that the convergence results are valid in a more general setting.

Assyr Abdulle, Ping Lin, Alexander V. Shapeev

In this paper we prove a priori and a posteriori error estimates for a multiscale numerical method for computing equilibria of multilattices under an external force. The error estimates are derived in a $W^{1,\infty}$ norm in one space dimension. One of the features of our analysis is that we establish an equivalent way of formulating the coarse-grained problem which greatly simplifies derivation of the error bounds (both, a priori and a posteriori). We illustrate our error estimates with numerical experiments.

Alexander V. Shapeev, Mitchell Luskin

The present paper aims at developing a theory of computation of crystalline defects at finite temperature. In a one-dimensional setting we introduce Gibbs distributions corresponding to such defects and rigorously establish their asymptotic expansion. We then give an example of using such asymptotic expansion to compare the accuracy of computations using the free boundary conditions and using an atomistic-to-continuum coupling method.

Timofei Miryashkin, Olga Klimanova, Alexander Shapeev

Conflicting experiments disagree on whether the titanium-vanadium (Ti-V) binary alloy exhibits a body-centred cubic (BCC) miscibility gap or remains completely soluble. A leading hypothesis attributes the miscibility gap to oxygen contamination during alloy preparation. To resolve this disagreement, we use an ab initio + machine-learning workflow that couples an actively-trained Moment Tensor Potential with Bayesian inference of free energy surface. This workflow enables construction of the Ti-V phase diagram across the full composition range with systematically reduced statistical and finite-size errors. The resulting diagram reproduces all experimental features, demonstrating the robustness of our approach, and clearly favors the variant with a BCC miscibility gap terminating at T = 980 K and c = 0.67. Because our simulations model a perfectly oxygen-free Ti-V system, the observed gap cannot originate from impurity effects, in contrast to recent CALPHAD reassessments.

A. A. Solovykh, N. E. Rybin, I. S. Novikov et al.

Accounting for nuclear quantum effects (NQEs) can significantly alter material properties at finite temperatures. Atomic modeling using the path-integral molecular dynamics (PIMD) method can fully account for such effects, but requires computationally efficient and accurate models of interatomic interactions. Empirical potentials are fast but may lack sufficient accuracy, whereas quantum-mechanical calculations are highly accurate but computationally expensive. Machine-learned interatomic potentials offer a solution to this challenge, providing near-quantum-mechanical accuracy while maintaining high computational efficiency compared to density functional theory (DFT) calculations. In this context, an interface was developed to integrate moment tensor potentials (MTPs) from the MLIP-2 software package into PIMD calculations using the i-PI software package. This interface was then applied to active learning of potentials and to investigate the influence of NQEs on material properties, namely the temperature dependence of lattice parameters and thermal expansion coefficients, as well as radial distribution functions, for lithium hydride (LiH) and silicon (Si) systems. The results were compared with experimental data, quasi-harmonic approximation calculations, and predictions from the universal machine learning force field MatterSim. These comparisons demonstrated the high accuracy and effectiveness of the MTP-PIMD approach.

Timofei Miryashkin, Olga Klimanova, Vladimir Ladygin et al.

Phase diagrams serve as a highly informative tool for materials design, encapsulating information about the phases that a material can manifest under specific conditions. In this work, we develop a method in which Bayesian inference is employed to combine thermodynamic data from molecular dynamics (MD), melting point simulations, and phonon calculations, process these data, and yield a temperature-concentration phase diagram. The employed Bayesian framework yields us not only the free energies of different phases as functions of temperature and concentration but also the uncertainties of these free energies originating from statistical errors inherent to finite-length MD trajectories. Furthermore, it extrapolates the results of the finite-atom calculations to the infinite-atom limit and facilitates the choice of temperature, chemical potentials, and the number of atoms conducting the next simulation with which will be the most efficient in reducing the uncertainty of the phase diagram. The developed algorithm was successfully tested on two binary systems, Ge-Si and K-Na, in the full range of concentrations and temperatures.

Evgenii Tsymbalov, Sergei Makarychev, Alexander Shapeev et al.

Active learning methods for neural networks are usually based on greedy criteria which ultimately give a single new design point for the evaluation. Such an approach requires either some heuristics to sample a batch of design points at one active learning iteration, or retraining the neural network after adding each data point, which is computationally inefficient. Moreover, uncertainty estimates for neural networks sometimes are overconfident for the points lying far from the training sample. In this work we propose to approximate Bayesian neural networks (BNN) by Gaussian processes, which allows us to update the uncertainty estimates of predictions efficiently without retraining the neural network, while avoiding overconfident uncertainty prediction for out-of-sample points. In a series of experiments on real-world data including large-scale problems of chemical and physical modeling, we show superiority of the proposed approach over the state-of-the-art methods.

Evgenii Tsymbalov, Maxim Panov, Alexander Shapeev

Active learning is relevant and challenging for high-dimensional regression models when the annotation of the samples is expensive. Yet most of the existing sampling methods cannot be applied to large-scale problems, consuming too much time for data processing. In this paper, we propose a fast active learning algorithm for regression, tailored for neural network models. It is based on uncertainty estimation from stochastic dropout output of the network. Experiments on both synthetic and real-world datasets show comparable or better performance (depending on the accuracy metric) as compared to the baselines. This approach can be generalized to other deep learning architectures. It can be used to systematically improve a machine-learning model as it offers a computationally efficient way of sampling additional data.

Alexander V. Shapeev

Density functional theory offers a very accurate way of computing materials properties from first principles. However, it is too expensive for modelling large-scale molecular systems whose properties are, in contrast, computed using interatomic potentials. The present paper considers, from a mathematical point of view, the problem of constructing interatomic potentials that approximate a given quantum-mechanical interaction model. In particular, a new class of systematically improvable potentials is proposed, analyzed, and tested on an existing quantum-mechanical database.

Derek Olson, Alexander V. Shapeev, Pavel Bochev et al.

We formulate and analyze an optimization-based Atomistic-to-Continuum (AtC) coupling method for problems with point defects. Near the defect core the method employs a potential-based atomistic model, which enables accurate simulation of the defect. Away from the core, where site energies become nearly independent of the lattice position, the method switches to a more efficient continuum model. The two models are merged by minimizing the mismatch of their states on an overlap region, subject to the atomistic and continuum force balance equations acting independently in their domains. We prove that the optimization problem is well-posed and establish error estimates.

Ping Lin, Alexander V. Shapeev

This paper studies numerical methods for accurate treatment of the interface between the local and the nonlocal region in a QC approximation of atomistic materials. Only the energy-based methods are considered. Particularly, a quasicontinuum projection (QCP) method based on the idea of finite elements is shown to be accurate and efficient for this problem. We analyse the QCP method and study its relation to the existing methods, such as the quasinonlocal quasicontinuum method and the geometrically consistent reconstruction-based method. The analysis and the results of numerical tests confirm that the projection-based QC method successfully removes the ghost force with the same computational cost as the other methods. In all computed examples the error of QCP is either the same or lower as the error of the other methods. The performance of these methods in treating interfaces of elements in the local region is also examined.