Model reduction and simulation at the atomistic scale

About this event

This is a one-day MIR@W workshop bringing together researchers in Applied Maths and Scientific Computation.

To register your interest in attending this event, please fill out the form here.

Financial support for this event comes from the Warwick-SJTU Seed Fund, the Warwick Mathematics Institute through the MIR@W day scheme, and the London Mathematical Society.

Event details and schedule

This event will take place on Friday 26 January 2024 in A0.28, Milburn HouseLink opens in a new window.

Important: To access the building you will need to use the north entrance (closest to both Kirby Corner Road and the room as indicated on the interactive map link above). This entrance is card-access only for Warwick staff and students, but someone will be available to let external participants in.

Schedule

10.00 - Welcome
10.15 - Operator Learning for Multiscale PDEs, Lei Zhang (SJTU)
10.45 - Dislocation dynamics parameterisation in Ni-based superalloys from atomistic simulations, Geraldine Anis (Warwick)
11.00 - Break
11.30 - Application of machine-learned interatomic potentials in atomic-scale simulations and beyond, Yangshuai Wang (UBC Vancouver)
12.00 - ODE-Net based data-driven modelling for crowd dynamics, Jinglai Li (Birmingham)
12.30 - Lunch (elsewhere on campus)
14.00 - Predicting diffusion and plasticity at the atomistic scale, Tom Swinburne (CINaM, Marseille)
14.30 - Atomistic modelling of nucleation - reduction to the committor, Kat Blow (Warwick)
14.45 - Structure-property relations for block copolymers - A molecular data-driven approach, Arre Rajkumar (Warwick)
15.00 - Break
15.30 - Accurate and efficient splitting methods for dissipative particle dynamics, Xiaocheng Shang (Birmingham)
16.00 - Close

Detailed talk abstracts can be found below.

Talk Abstracts

Operator Learning for Multiscale PDEs, Lei Zhang

Multiscale partial differential equations (MsPDEs) play a crucial role in applications involving heterogeneous and random media, enabling the prediction of complex phenomena such as reservoir modeling, atmospheric and ocean circulation, and high-frequency scattering. On the other hand, neural operators [1] have emerged as powerful tools for learning the mapping between the infinite-dimensional parameter and solution spaces of PDEs. However, when it comes to multiscale PDEs, the spectral bias [1,2] towards low-frequency components poses a significant obstacle for existing neural operators. To tackle this challenge, we propose models based on hierarchical attention and convolution [4,5], inspired by the hierarchical matrix approach. Our model incorporates a scale-adaptive interaction range and self-attentions across multiple levels, allowing nested feature computation with controllable linear cost and the encoding/decoding of the multiscale solution space. Additionally, we introduce an empirical H^1 loss function to enhance the learning of high-frequency components. Through numerical experiments, we demonstrate that our approach outperforms state-of-the-art methods for representative multiscale problems.

[1] Zongyi Li, Nikola Kovachki, Kamyar Azizzadenesheli, Burigede Liu, Kaushik Bhattacharya, Andrew Stuart, and Anima Anandkumar, ICLR 2021

[2] N. Rahaman, D. Arpit, A. Baratin, F. Draxler, M. Lin, F. A. Hamprecht, Y. Bengio, A. Courville, On the spectral bias of deep neural networks, ICML, 2019

[3] Zhiqin Xu, Yaoyu Zhang, Tao Luo, Yanyang Xiao, Zheng Ma, Frequency principle: Fourier analysis sheds light on Deep Neural Networks, Communications in Computational Physics 28 (5), 1746จC1767, 27, 2020

[4] Xinliang Liu, Bo Xu, Shuhao Cao, Lei Zhang, HT-Net: Hierarchical Transformer based Operator Learning Model for Multiscale PDEs. arXiv:2210.10890, submitted.

[5] Bo Xu, Xinliang Liu, Lei Zhang, Dilated convolution neural operator for multiscale partial differential equations, in preparation.

Dislocation dynamics parameterisation in Ni-based superalloys from atomistic simulations, Geraldine Anis

Ni-based superalloys are important materials for high temperature applications. Nanoscale precipitates in their microstructure hinder dislocation motion, which results in the extraordinary strengthening effect at elevated temperatures. In our work, we study the deformation behaviour of these materials by observing and parameterising the motion of edge dislocations from Molecular Dynamics (MD) simulations using classical effective potentials. Pure
face-cantered cubic (FCC) Ni is used as a model system to represent an idealised γ matrix solid solution phase in Ni-based superalloys. Differential Evolution Monte Carlo (DE-MC) is applied within a Bayesian framework to fit the parameters of an equation of motion to the dislocation trajectories extracted from the MD simulations. The parameters of interest are the effective mass, drag coefficient, and force experienced by the dislocation as the simulation
cell is sheared and the dislocation moves. Parameterising dislocation motion in this way means that the fitting procedure yields physically meaningful parameters, which can then be used as inputs to larger lengthscale simulations. Moreover, using DE-MC sampling makes it possible to obtain parameter distributions and propagate uncertainties through the model, quantifying the uncertainty in its predictions: the dislocation positions and velocities. This work serves as a first step towards building a more comprehensive surrogate model that can describe the deformation behaviour of Ni-based superalloys. This reduced model should in future also account for dislocation-dislocation and dislocation-precipitate interactions for a more realistic representation of Ni-based superalloys.

Application of machine-learned interatomic potentials in atomic-scale simulations and beyond, Yangshuai Wang

This presentation will explore recent advancements in the field of machine-learned interatomic potentials (MLIPs), their impact on molecular simulations, and their potential for enhancing our understanding of materials behavior at the atomic scale and beyond. Starting with an overview of the Atomic Cluster Expansion (ACE) method, we will delve into its construction, recent advancements, and its role in molecular dynamics simulations. We will then discuss the generalization error analysis of ACE for simulating crystalline defects, focusing on defect equilibrium studies. Furthermore, we will explore various applications of MLIPs in materials science at the atomic scale. Finally, we will introduce MLIPs-assisted coarse-grained molecular dynamics, a method that extends simulation scales in both temporal and spatial domains using the state-of-the-art ACE method.

ODE-Net based data-driven modelling for crowd dynamics, Jinglai Li

Predicting the behaviors of pedestrian crowds is of critical importance for a variety of real-world problems. Data driven modeling, which aims to learn the mathematical models from observed data, is a promising tool to construct models that can make accurate predictions of such systems. In this work, we present a data-driven modeling approach based on the ODE-Net framework, for constructing continuous-time models of crowd dynamics. We discuss some challenging issues in applying the ODE-Net method to such problems, which are primarily associated with the dimensionality of the underlying crowd system, and we propose to address these issues by incorporating the social-force concept in the ODE-Net framework. Finally application examples are provided to demonstrate the performance of the proposed method.

Predicting diffusion and plasticity at the atomistic scale, Tom Swinburne

I will discuss two methods to coarse-grain, analyse and predict molecular dynamics simulations of materials, focusing on solid-state diffusion and plasticity.

Solid state diffusion is typically metastable (thermally activated). In this regime, kinetics can be rigorously mapped to a discrete Markov chain with robust Bayesian bounds on the influence of as-yet unseen transitions, which enables the future of jump trajectories in the observed state space to be predicted. These bounds enable resource allocation to be optimised in massively parallel deployment, and can be propagated to measure the convergence of coarse-grained transport properties [1].

However, in many settings of interest, such as plastic deformation, atomic kinetics show no clear signature of metastabilty and atomic structures cannot be meaningfully represented by low-dimensional collective variables. I will argue many-body “descriptor" functions used in atomic machine learning form an ideal “medium-dimensional” latent space for coarse-graining and analysis. A vector autoregressive model can resample and forecast descriptor trajectories using a Mahalanobis distance to qualify forecast uncertainty, with the descriptor manifold clearly sensitive to structural transitions [2].

[1] TD Swinburne and D Perez, NPJ Computational Materials 2020

[2] TD Swinburne, Physical Review Letters, 2023

Atomistic modelling of nucleation - reduction to the committor, Kat Blow

Nucleation is an important and widespread phenomenon impacting climate modelling and many areas of industry. Computer simulations provide an avenue to explore these problems but require the use of order parameters to distinguish between liquid and solid configurations - the ideal mapping is known as the committor. In this talk, order parameters in the context of simulations of nucleation are introduced, and some of the difficulties of using these are discussed.

Structure-property relations for block copolymers - A molecular data-driven approach, Arre Rajkumar

Block copolymers possess the ability to self-assemble in to a surprisingly diverse variety of ordered mesoscopic structures, which themselves possess interesting physical properties. The first part of this talk will be dedicated to the molecular structure of block copolymers. We employ Monte Carlo simulation to generate molecular configurations of self-assembled block copolymers by first predicting the self-assembled morphology via self-consistent field theory calculations. This in turn provides an extensible methodology to simulate a wide variety of self-assembled block copolymers at the microscale, a task which until now was limited only to the simplest of copolymer systems. The second part of the talk is then dedicated to property. After using molecular dynamics in conjunction with the aforementioned Monte Carlo method to generate trajectory data, the use of statistical inference or machine learning is employed as a means of building constitutive relations for block copolymers. Specifically, we employ Gaussian process regression as a means of modelling the structure-property relations of block copolymers. These constitutive relations are entirely informed by molecular scale data and demonstrate a methodology to bridge the molecular scale and the macroscopic scale, replacing expensive traditional modelling methods with a cost-effective, data-driven approach.

Accurate and efficient splitting methods for dissipative particle dynamics, Xiaocheng Shang

We study numerical methods for dissipative particle dynamics (DPD), which is a system of stochastic differential equations and a popular stochastic momentum-conserving thermostat for simulating complex hydrodynamic behaviour at mesoscales. We propose a new splitting method that is able to substantially improve the accuracy and efficiency of DPD simulations in a wide range of friction coefficients, particularly in the extremely large friction limit that corresponds to a fluid-like Schmidt number, a key issue in DPD. Various numerical experiments on both equilibrium and transport properties are performed to demonstrate the superiority of the newly proposed method over popular alternative schemes in the literature.