Mathematical Aspects of Atomistic and Multiscale Modelling

材料的基本构件是组成它们的原子，这些原子通过它们的相关电子的适当排列在彼此之间形成键。材料变形和破坏（例如涡轮叶片的破裂）影响宏观尺度，但在原子和电子尺度上开始和传播，这种耦合在传统（连续介质力学）模型中很大程度上被忽略了。正在进行的对材料失效和许多其他宏观过程的预测性描述的研究越来越多地集中在原子尺度上，并且需要考虑原子和电子结构以及宏观信息（如应变）的计算模型。电子结构的量子力学模型为原子之间的作用力提供了精确的、可转移的（即广泛适用的）模型，但它们的极端计算成本使它们不适合模拟复杂的原子过程，如可塑性、开裂或化学反应，这些过程跨越许多长度和时间尺度。研究原子机制的典型替代方案是原子间势（或经验力场），这是精度较差的经验模型，并且在应用程序之间的可转移性极其有限。原子间作用力的精确但计算效率仍然很高的模型将立即在广泛的学科中得到应用，特别是材料科学和生物化学。从2010年开始，一个令人兴奋的发展是通过通用逼近器（机器学习）构建原子间势；也就是说，把原子间势的构造看作近似问题而不是建模问题。这种范式转变为数学理论形式化这个极其丰富的近似问题创造了机会，并支持和加速了下一步的创新步骤。本计划的目的是从应用数学的角度，特别是应用分析、近似理论和数值分析的角度来探讨这一问题。数学技术将占据中心位置，但也将整合和综合物理学、化学和数据科学的思想，以建立一种有纪律的方法来支持并在适当的情况下领导下一代原子间势的设计。科学和工业的影响将通过与科学团体的密切合作以及通过软件共同开发来实现。从长远来看，这项研究将扩展到其他原子建模场景中类似的粗粒度挑战。例如，许多技术可以转移到开发新的紧密绑定模型、经典DFT模型或粗粒度动态系统。它们也可以用于分析原子结构数据库，以加速，例如，在药物和材料设计中应用的结构-性质关系的自动发现。

The basic building blocks of materials are their constituent atoms which create bonds between one another via suitable arrangements of their associated electrons. Material deformation and failure (e.g., cracking of a turbine blade) affects macro-scopic scales but initiates and propagates at the scale of atoms and electrons, a coupling which is largely ignored in conventional (continuum mechanics) models. The ongoing search for predictive descriptions of material failure and many other macroscopic processes increasingly focuses on the atomic scale and requires computational models that take into account both atomistic and electronic structure as well as macroscopic information such as strain. Quantum mechanical models of electronic structure provide accurate and transferable (i.e., widely applicable) models for forces acting between atoms, but their extreme computational cost makes them unsuitable for simulating complex atomistic processes such as plasticity, cracking or chemical reactions, that span many length- and time-scales. The canonical alternative to study atomistic mechanisms are interatomic potentials (or, empirical force fields), which are empirical models with poor accuracy and extremely limited transferability across applications. Accurate but still computationally efficient models for interatomic forces would immediately have applications across a wide range of disciplines, in particular materials science and bio-chemistry. An exciting development, starting ca 2010, is to build interatomic potentials from universal approximators (machine learning); that is, to treat the construction of interatomic potentials as an approximation problem instead of a modelling problem. This paradigm shift creates an opportunity for a mathematical theory to formalise this extremely rich approximation problem and support and accelerate the next innovation steps. The aim of the proposed program is to explore this problem from the perspective of applied mathematics, in particular applied analysis, approximation theory and numerical analysis. Mathematical techniques will take center-stage but incorporate and synthesize ideas from physics, chemistry and data-science to establish a disciplined approach to support and where appropriate lead the design of the next generation of interatomic potentials. Impact in the sciences and industry will be achieved through close collaboration with science groups and through software co-development. In the long term the research will broaden into similar coarse-graining challenges in other atomistic modelling scenarios. For example, many techniques will be transferrable to developing new tight-binding models, classical DFT models, or coarse-grained dynamical systems. They can also be adapted to analyze databases of atomic structures to accelerate, e.g., automated discovery of structure-property relations with applications in drug and material design.