Mathematical Aspects of Atomistic and Multiscale Modelling

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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.