arXiv subjects

Yangshuai Wang

Publications and source records attributed to Yangshuai Wang.

46 records · Page 3Linked to original sources

arXiv2023

A Theoretical Case Study of the Generalisation of Machine-learned Potentials

Machine-learned interatomic potentials (MLIPs) are typically trained on datasets that encompass a restricted subset of possible input structures, which presents a potential challenge for their generalization to a broader range of systems outside the training set. Nevertheless, MLIPs have demonstrated impressive accuracy in predicting forces and energies in simulations involving intricate and complex structures. In this paper we aim to take steps towards rigorously explaining the excellent observed generalisation properties of MLIPs. Specifically, we offer a comprehensive theoretical and numerical investigation of the generalization of MLIPs in the context of dislocation simulations. We quantify precisely how the accuracy of such simulations is directly determined by a few key factors: the size of the training structures, the choice of training observations (e.g., energies, forces, virials), and the level of accuracy achieved in the fitting process. Notably, our study reveals the crucial role of fitting virials in ensuring the consistency of MLIPs for dislocation simulations. Our series of careful numerical experiments encompassing screw, edge, and mixed dislocations, supports existing best practices in the MLIPs literature but also provides new insights into the design of data sets and loss functions.

physics.comp-ph↗

arXiv2023

A Posteriori Analysis and Adaptive Algorithms for Blended Type Atomistic-to-Continuum Coupling with Higher-Order Finite Elements

The efficient and accurate simulation of material systems with defects using atomistic- to-continuum (a/c) coupling methods is a topic of considerable interest in the field of computational materials science. To achieve the desired balance between accuracy and computational efficiency, the use of a posteriori analysis and adaptive algorithms is critical. In this work, we present a rigorous a posteriori error analysis for three typical blended a/c coupling methods: the blended energy-based quasi-continuum (BQCE) method, the blended force-based quasi-continuum (BQCF) method, and the atomistic/continuum blending with ghost force correction (BGFC) method. We employ first and second-order finite element methods (and potentially higher-order methods) to discretize the Cauchy-Born model in the continuum region. The resulting error estimator provides both an upper bound on the true approximation error and a lower bound up to a theory-based truncation indicator, ensuring its reliability and efficiency. Moreover, we propose an a posteriori analysis for the energy error. We have designed and implemented a corresponding adaptive mesh refinement algorithm for two typical examples of crystalline defects. In both numerical experiments, we observe optimal convergence rates with respect to degrees of freedom when compared to a priori error estimates.

math.NA↗

arXiv2022

Efficient a Posteriori Error Control of a Consistent Atomistic/Continuum Coupling Method for Two Dimensional Crystalline Defects

Adaptive atomistic/continuum (a/c) coupling method is an important method for the simulation of material and atomistic systems with defects to achieve the balance of accuracy and efficiency. Residual based a posteriori error estimator is often employed in the adaptive algorithm to provide an estimate of the error of the strain committed by applying the continuum approximation for the atomistic system and the finite element discretization in the continuum region. In this work, we propose a theory based approximation for the residual based a posteriori error estimator which greatly improves the efficiency of the adaptivity. In particular, the numerically expensive modeling residual is only computed exactly in a small region around the coupling interface but replaced by a theoretically justified approximation by the coarsening residual outside that region. We present a range of adaptive computations based on our modified a posteriori error estimator and its variants for different types of crystalline defects some of which are not considered in previous related literature of the adaptive a/c methods. The numerical results show that, compared with the original residual based error estimator, the adaptive algorithm using the modified error estimator with properly chosen parameters leads to the same optimal convergence rate of the error but reduces the computational cost by one order with respect to the number of degrees of freedom.

math.NA↗

arXiv2022

A framework for a generalisation analysis of machine-learned interatomic potentials

Machine-learned interatomic potentials (MLIPs) and force fields (i.e. interaction laws for atoms and molecules) are typically trained on limited data-sets that cover only a very small section of the full space of possible input structures. MLIPs are nevertheless capable of making accurate predictions of forces and energies in simulations involving (seemingly) much more complex structures. In this article we propose a framework within which this kind of generalisation can be rigorously understood. As a prototypical example, we apply the framework to the case of simulating point defects in a crystalline solid. Here, we demonstrate how the accuracy of the simulation depends explicitly on the size of the training structures, on the kind of observations (e.g., energies, forces, force constants, virials) to which the model has been fitted, and on the fit accuracy. The new theoretical insights we gain partially justify current best practices in the MLIP literature and in addition suggest a new approach to the collection of training data and the design of loss functions.

math.NA↗

arXiv2022

Adaptive Multigrid Strategy for Geometry Optimization of Large-Scale Three Dimensional Molecular Mechanics

In this paper, we present an efficient adaptive multigrid strategy for the geometry optimization of large-scale three dimensional molecular mechanics. The resulting method can achieve significantly reduced complexity by exploiting the intrinsic low-rank property of the material configurations and by combining the state-of-the-art adaptive techniques with the hierarchical structure of multigrid algorithms. To be more precise, we develop a oneway multigrid method with adaptive atomistic/continuum (a/c) coupling, e.g., blended ghost force correction (BGFC) approximations with gradient-based a posteriori error estimators on the coarse levels. We utilize state-of-the-art 3D mesh generation techniques to effectively implement the method. For 3D crystalline defects, such as vacancies, micro-cracks and dislocations, compared with brute-force optimization, complexity with superior rates can be observed numerically, and the strategy has a five-fold acceleration in terms of CPU time for systems with $10^8$ atoms.

physics.comp-ph↗

arXiv2022

Theoretical Study of Elastic Far-Field Decay from Dislocations in Multilattices

We precisely and rigorously characterise the decay of elastic fields generated by dislocations in crystalline materials, focusing specifically on the role of multilattices. Concretely, we establish that the elastic field generated by a dislocation in a multilattice can be decomposed into a continuum field predicted by a linearised Cauchy-Born elasticity theory, and a discrete and nonlinear core corrector representing the defect core. We demonstrate both analytically and numerically the consequences of this result for cell size effects in numerical simulations.

math.AP↗

arXiv2021

QM/MM Methods for Crystalline Defects. Part 3: Machine-Learned Interatomic Potentials

We develop and analyze a framework for consistent QM/MM (quantum/classic) hybrid models of crystalline defects, which admits general atomistic interactions including traditional off-the-shell interatomic potentials as well as state of art "machine-learned interatomic potentials". We (i) establish an a priori error estimate for the QM/MM approximations in terms of matching conditions between the MM and QM models, and (ii) demonstrate how to use these matching conditions to construct practical machine learned MM potentials specifically for QM/MM simulations.

math.NA↗

arXiv2020

A Posteriori Error Estimates for Adaptive QM/MM Coupling Methods

Hybrid quantum/molecular mechanics models (QM/MM methods) are widely used in material and molecular simulations when MM models do not provide sufficient accuracy but pure QM models are computationally prohibitive. Adaptive QM/MM coupling methods feature on-the-fly classification of atoms during the simulation, allowing the QM and MM subsystems to be updated as needed. In this work, we propose such an adaptive QM/MM method for material defect simulations based on a new residual based it a posteriori error estimator, which provides both lower and upper bounds for the true error. We validate the analysis and illustrate the effectiveness of the new scheme on numerical simulations for material defects.

math.NA↗

arXiv2019

A Priori Analysis of a Higher Order Nonlinear Elasticity Model for an Atomistic Chain with Periodic Boundary Condition

Nonlinear elastic models are widely used to describe the elastic response of crystalline solids, for example, the well-known Cauchy-Born model. While the Cauchy-Born model only depends on the strain, effects of higher order strain gradients are significant and higher order continuum models are preferred, in various applications such as defect dynamics and modeling of carbon nanotubes. In this paper, we rigorously derive a higher order nonlinear elasticity model for crystals from its atomistic description in one dimension. We show that, compared to the second order accuracy of the Cauchy-Born model, the higher order continuum model in this paper is of fourth oder accuracy with respect to the interatomic spacing in the thermal dynamic limit. In addition, we discuss the key issues for the derivation of higher order continuum models in more general cases. The theoretical convergence results are demonstrated by numerical experiments.

math.NA↗

arXiv2018

Adaptive QM/MM Coupling for Crystalline Defects

QM (quantum mechenics) and MM (molecular mechenics) coupling methods are widely used in simulations of crystalline defects. In this paper, we construct a residual based a posteriori error indicator for QM/MM coupling approximations. We prove the reliability of the error indicator (upper bound of the true approximation error) and develop some sampling techniques for its efficient calculation. Based on the error indicator and Dörfler marking strategy, we design an adaptive QM/MM algorithm for crystalline defects and demonstrate the efficiency with some numerical experiments.

math.NA↗