Partial Differential Equations (PDE) are fundamental to model different phenomena in science and engineering mathematically. Solving them is a crucial step towards a precise knowledge of the behaviour of natural and engineered systems. In general, in order to solve PDEs that represent real systems to an acceptable degree, analytical methods are usually not enough. One has to resort to discretization methods. For engineering problems, probably the best known option is the finite element method (FEM). However, powerful alternatives such as mesh-free methods and Isogeometric Analysis (IGA) are also available. The fundamental idea is to approximate the solution of the PDE by means of functions specifically built to have some desirable properties. In this contribution, we explore Deep Neural Networks (DNNs) as an option for approximation. They have shown impressive results in areas such as visual recognition. DNNs are regarded here as function approximation machines. There is great flexibility to define their structure and important advances in the architecture and the efficiency of the algorithms to implement them make DNNs a very interesting alternative to approximate the solution of a PDE. We concentrate in applications that have an interest for Computational Mechanics. Most contributions that have decided to explore this possibility have adopted a collocation strategy. In this contribution, we concentrate in mechanical problems and analyze the energetic format of the PDE. The energy of a mechanical system seems to be the natural loss function for a machine learning method to approach a mechanical problem. As proofs of concept, we deal with several problems and explore the capabilities of the method for applications in engineering.
Density functional theory calculations are robust tools to explore the mechanical properties of pristine structures at their ground state but become exceedingly expensive for large systems at finite temperatures. Classical molecular dynamics (CMD) simulations offer the possibility to study larger systems at elevated temperatures, but they require accurate interatomic potentials. Herein the authors propose the concept of first‐principles multiscale modeling of mechanical properties, where ab initio level of accuracy is hierarchically bridged to explore the mechanical/failure response of macroscopic systems. It is demonstrated that machine‐learning interatomic potentials (MLIPs) fitted to ab initio datasets play a pivotal role in achieving this goal. To practically illustrate this novel possibility, the mechanical/failure response of graphene/borophene coplanar heterostructures is examined. It is shown that MLIPs conveniently outperform popular CMD models for graphene and borophene and they can evaluate the mechanical properties of pristine and heterostructure phases at room temperature. Based on the information provided by the MLIP‐based CMD, continuum models of heterostructures using the finite element method can be constructed. The study highlights that MLIPs were the missing block for conducting first‐principles multiscale modeling, and their employment empowers a straightforward route to bridge ab initio level accuracy and flexibility to explore the mechanical/failure response of nanostructures at continuum scale.
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