Machine learning materials physics: Multi-resolution neural networks learn the free energy and nonlinear elastic response of evolving microstructures

Abstract

Many important multi-component crystalline solids undergo mechanochemical spinodal decomposition: a phase transformation in which the compositional redistribution is coupled with structural changes of the crystal, resulting in dynamic and intricate microstructures. The ability to rapidly compute the macroscopic behavior based on these detailed microstructures is of paramount importance for accelerating material discovery and design. However, the evaluation of macroscopic, nonlinear elastic properties purely based on direct numerical simulations (DNS) is computationally very expensive, and hence impractical for material design when a large number of microstructures need to be tested. A further complexity of a hierarchical nature arises if the elastic free energy and its variation with strain is a small scale fluctuation on the dominant trajectory of the total free energy driven by microstructural dynamics. To address these challenges, we present a data-driven approach, which combines advanced neural network (NN) models with DNS to predict the mechanical free energy and homogenized stress fields on microstructures in a family of two-dimensional multi-component crystalline solids. The microstructres are numerically generated by solving a coupled, Cahn-Hilliard and nonlinear strain gradient elasticity problem. The hierarchical structure of the free energy's evolution induces a multi-resolution character to the machine learning paradigm: We construct knowledge-based neural networks (KBNNs) with either pre-trained fully connected deep neural networks (DNNs) or pre-trained convolutional neural networks (CNNs) that describe the dominant feature of the data to fully represent the hierarchichally evolving free energy. We demonstrate multi-resolution learning of the materials physics of nonlinear elastic response for both fixed and evolving microstructures. Keywords deep neural networks • convolutional neural networks • knowledge-based neural networks • mechanochemical spinodal decomposition • homogenization • mechanical free energy

Exploring the potential of transfer learning for metamodels of heterogeneous material deformation

“…Conceptually, multi-fidelity models are a form of transfer learning. Notably, many multi-fidelity modeling methods presented in the literature involve specialized model architecture and/or problem specific methods for data integration [55]. In this work, our goal is to explore the efficacy of transfer learning exclusively through straightforward model pretraining.…”

Learning Constitutive Relations using Symmetric Positive Definite Neural Networks

“…Additionally, neural networks are used to address more complex material behaviors, such as microcracking, brittle fracture, and crack propagation [9,10,11,12]. Moreover, attempts for new material designs are made by leveraging neural networks with microscopic structure parameters incorporated in the inputs [13,14,15,16]. The cited literature shows the potential capability of neural networks to represent complex constitutive relations.…”

System inference for the spatio-temporal evolution of infectious diseases: Michigan in the time of COVID-19

“…During the COVID-19 Pandemic, the widespread availability of data in the public domain [6][7][8][9][10][11] has served to attract methods of mathematics, computation and data science to analyzing this information, inferring the disease's dynamics and making projections. The present communication is in this spirit, and brings our recent work in large scale computations of partial differential equations (PDEs), system inference and machine learning to this problem [12][13][14][15][16][17].…”

“…To these tasks we have brought the abundance of high-quality, public domain, data on the evolution of the various compartment pertaining to the SIRD model in the US state of Michigan. The temporal resolution by days and spatial resolution by the 85 counties of Michigan has allowed us to apply our methods of Variational System Identification [12,13], PDE-constrained optimization and machine learning [14][15][16][17] to these data.…”