Open Access Article

This Open Access Article is licensed under a Creative Commons Attribution-Non Commercial 3.0 Unported Licence

Kai
Guo
^{a},
Zhenze
Yang
^{ab},
Chi-Hua
Yu
^{ac} and
Markus J.
Buehler
*^{ade}
^{a}Laboratory for Atomistic and Molecular Mechanics (LAMM), Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, 77 Massachusetts Ave. 1-290, Cambridge, Massachusetts 02139, USA. E-mail: mbuehler@MIT.EDU; Tel: +1 617 452 2750
^{b}Department of Materials Science and Engineering, Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, Massachusetts 02139, USA
^{c}Department of Engineering Science, National Cheng Kung University, No. 1, University Road, Tainan City 701, Taiwan
^{d}Center for Computational Science and Engineering, Schwarzman College of Computing, Massachusetts Institute of Technology, 77 Massachusetts Ave, Cambridge, Massachusetts 02139, USA
^{e}Center for Materials Science and Engineering, 77 Massachusetts Ave, Cambridge, Massachusetts 02139, USA

Received
7th September 2020
, Accepted 16th December 2020

First published on 17th December 2020

Artificial intelligence, especially machine learning (ML) and deep learning (DL) algorithms, is becoming an important tool in the fields of materials and mechanical engineering, attributed to its power to predict materials properties, design de novo materials and discover new mechanisms beyond intuitions. As the structural complexity of novel materials soars, the material design problem to optimize mechanical behaviors can involve massive design spaces that are intractable for conventional methods. Addressing this challenge, ML models trained from large material datasets that relate structure, properties and function at multiple hierarchical levels have offered new avenues for fast exploration of the design spaces. The performance of a ML-based materials design approach relies on the collection or generation of a large dataset that is properly preprocessed using the domain knowledge of materials science underlying chemical and physical concepts, and a suitable selection of the applied ML model. Recent breakthroughs in ML techniques have created vast opportunities for not only overcoming long-standing mechanics problems but also for developing unprecedented materials design strategies. In this review, we first present a brief introduction of state-of-the-art ML models, algorithms and structures. Then, we discuss the importance of data collection, generation and preprocessing. The applications in mechanical property prediction, materials design and computational methods using ML-based approaches are summarized, followed by perspectives on opportunities and open challenges in this emerging and exciting field.

Over the past a few decades, it has been found that artificial intelligence (AI), a study of computations which perceive, reason, and act like human beings, has the potential to address these challenges.^{15} Specifically, the most promising one is an approach to AI called machine learning (ML), which can discover the mapping from high-throughput input data to output that is used to make decisions. In simple ML algorithms, the representation of input data is hand-designed by researchers, and each piece in the representation is referred to as a feature. Yet, it was extremely challenging to manually extract appropriate features from some sort of raw data that are easy to understand for human but difficult for machines, i.e., photographs of streets where cars are supposed to be recognized, until the emerge of deep learning (DL), a specific type of ML that can not only learn the representation of the input data but also parse the representation into multiple levels—from simple features to abstract ones—attributed to complex neural network structures.^{16} ML, especially DL, has achieved many exciting breakthroughs in algorithms and led to great success in computer vision, natural language processing and autonomous driving.^{17} Materials and mechanics communities are aware of the great opportunities of leveraging ML as a potential new paradigm. Several general reviews of materials design using ML have been published during the past few years.^{18–21} In the meantime, numerous research articles in this topic are coming out, and so do reviews of ML in specific materials or mechanics branches, involving energy materials,^{22,23} glasses,^{24} composites,^{25} polymers,^{26} bio-inspired materials,^{27} additive manufacturing,^{28,29} continuum materials mechanics,^{30} and so on.

In this review, we focus on reviewing the growth and state of the art of research efforts on mechanical materials design using ML, and also attempt to depict a general methodology for performing ML-based mechanical materials researches. As schematically shown in Fig. 1, a typical workflow for combining ML and materials research consists of three key components: (i) a well-organized material dataset either collected from literature and existing databases or generated from experiments and simulations; (ii) a ML model that is capable to learn and parse the representation for certain tasks; and (iii) a well-defined research problem of mechanical materials that has not been addressed by conventional methods, or has been solved but can be outperformed by ML-based approaches. A ML-based material research needs to glue all of these three components together, and a crucial step is the preprocessing of the raw material database into an appropriate numerical representation, also referred to as a descriptor. The preprocessed data should match the input data structure required by the selected ML model, and consist of essential material features to ensure high accuracy and training efficiency. A high-quality preprocessing requires not only expertise in mechanics and materials science, but also domain knowledge in related ML models. The former tells how to identify a challenging mechanical materials problem, acquire a database, and devise data preprocessing. The latter helps to select a suitable ML model to leverage and maximize its strength in given tasks, from prediction of mechanical behaviors of target materials, design of de novo mechanical materials, to development of new computational approaches.

To further discuss the foregoing methodology with the aid of present works in the literature, the paper is organized as follows. We begin with a brief summary of state-of-the-art ML models, algorithms and architectures. Readers can skip the description of the methods if they have already been familiar with them. To learn more about the methods of interest, we refer to the research articles and reviews cited in this section in which more details about the algorithms and examples are presented. Then we move on to a discussion of approaches to collect or generate datasets that are amenable to the ML models, followed by a review of existing applications of ML methods to various mechanical materials design problems. In these sections, inspiring strategies for data preparation, preprocessing, materials problem and ML model selection are highlighted. The paper is concluded with a few perspectives on the new computational paradigm that integrates mechanics and materials science with ML techniques.

Within this context, the simplest forms of ML without complex multilayer structures are classical ML algorithms. Linear regression (LIR)^{31} is one of the simplest algorithms aimed to find a linear relation between the input features and continuous output. Least Absolute Shrinkage and Selection Operator (LASSO)^{32} is a modification of LIR with additional absolute value penalization added to the loss function. Another reasonable extension of LIR is polynomial regression (PR)^{31} which includes polynomial terms in finding linear solutions. To further support non linearity, regression algorithms such as support vector regression (SVR)^{33} and random forest (RF)^{34} are introduced. These nonlinear models usually handle outliers better and show higher accuracy than linear models. Apart from regression, the other major category of ML tasks is classification. Instead of predicting specific values such as housing prices, the classification algorithms classify input into predefined categories. An example of classification algorithms is logistic regression (LOR),^{35} which is a classification algorithm with a loss function in logistic form despite it is named with “regression”. There are many other classical ML algorithms which can handle both regression and classification problems such as decision tree (DT)^{36} and gradient boosting.^{37,38}

Beyond classical ML techniques, scientists have developed artificial neural networks (ANNs), loosely inspired by the interconnected neurons in human brains, for deep data mining. The original idea is derived from perceptron, a simple precursor formulation dating back to 1958.^{39} By stacking multiple layers of neurons, a network structure is developed to learn nonlinear relation between input and output or delicate data distribution. As the depth of layer-by-layer networks increases, the resulting DL models offer tremendous impacts in computer science and various related interdisciplinary areas.

Feedforward neural networks (FFNNs) or multilayer perceptron (MLP)^{40,41} are probably the simplest and quintessential DL models. As the names indicate, the information passes through the network in a unidirectional manner for FFNNs. More specifically, each layer which consists of multiple neurons computes the output to the next layer based on the input from the previous layer. The weights or trainable parameters used for calculation for each neuron are optimized to minimize the loss function. In order to approach the minimum of the loss function during the training process, back propagation (BP), a widely used technique in ANNs training, is implemented together with gradient descent (GD) algorithm.^{42} BP functions as similar as calculating derivatives and GD algorithms determine the direction to jump down to the minimum. The process iterates until the loss function is close to its minimum.

Besides general FFNNs, two types of DL architectures are gaining vast attention due to their applications in computer vision and natural language processing (NLP), known as convolutional neural networks and recurrent neural networks.

Convolutional neural networks (CNNs) were first introduced in 1980,^{43} and reformulated in 1999.^{44} CNNs are image-based DL architecture by calculating mathematical operation “convolution” to extract features of images. Convolution preserves the spatial relationship between pixels and is calculated by multiplying the image matrix with the filter matrix. Filters contain trainable weights which are optimized during training for feature extraction. With different filters, separate operations such as edge detection can be performed to one image. By stacking the convolutional layers, simple features will be gradually assembled to intact and complicated ones.^{45} The CNNs are applied to and show exciting performances in face recognition, images classification and object detection.^{40} In materials design problems, with the capacity of capturing features at different hierarchical levels, CNNs are well suited to describe the properties of materials (which innately have hierarchical levels), especially biomaterials. These hierarchical features are not just found in materials, but in many other representations of matter, sound and language, and hence universal to the description of key societal systems.^{46,47}

Recurrent neural networks (RNNs) also gain popularity due to their capability of dealing with sequential data. In CNNs, inputs and outputs are supposed to be independent of each other, which might not be suitable for some tasks that emphasize the sequence of the data. For instance, given an incomplete sentence, it would be difficult to predict the next word if the sequential structure of the sentence is omitted. Instead, RNNs act on the sequential data with the output being depended on the previous and later sequence and utilize “memories” in determining output of each layer or state. For RNNs with large depth, the gradient calculated by BP easily vanishes or explodes.^{48,49} To address this issue, plenty of mechanisms including Long short-term memory (LSTM),^{50} Gated recurrent unit (GRU),^{51} ResNet^{52} and Attention^{53,54} have been developed, increasing the impact of RNNs in NLP tasks such as language translation and speech processing. RNNs also shed light on scientific problems such as protein folding and de novo protein design.^{55–57}

Generative models have been established to generate new data points based on the distribution of existing data. An intriguing and successful category of architectures among them are generative adversarial networks (GANs),^{58} which consist of two neural networks, the generator and the discriminator. The generator proposes new data instances and the discriminator compares the generated data with the real data. These two components contest with each other during the training as the generator aims to “fool” the discriminator by producing more genuine images while the discriminator attempts to distinguish real images from false images as accurately as possible. GANs reach convergence when the generator and the discriminator are at Nash equilibrium. The process of balancing the performances of the generator and the discriminator is somewhat similar to equilibrating a physical system with both attractive and repulsive forces which indicates that GANs can potentially shed light on describing physical phenomena. Furthermore, with the objective of generating fake data with restricted conditions or characteristics, a subtype of GANs named conditional GANs (cGANs)^{59} have been developed which include labels as a control variable. One of the applications of cGANs is image-to-image translation^{60,61} in which an image is used as the constrain of the generator. Unlike GANs, variational autoencoder (VAE)^{62} is another type of generative models that uses one neural network which first encodes the input data into an inexplainable code named as latent code and then decodes the latent code to reconstruct the output.

ML methods can also be used to evaluate and improve the performance of other applied ML models. Bayesian learning (BL)^{40} is an approach used for parameter estimation and probability comparison to evaluate a given algorithm. Gaussian process regression (GPR)^{63} is a nonparametric approach which can provide uncertainty measurements of predictions and build reduced-order models based on Bayesian learning. These approaches are potentially useful for mechanical materials designs problems as they are suitable for relatively small datasets and are working well without prior knowledge of model forms. Moreover, active learning is a learning algorithm that interactively inquires the user and selects data to be labeled.^{64} Training data would be augmented in an active learning loop with post-hoc experiments or simulations. For further discussion on the application of active learning in materials science, we refer to a recent review paper.^{65}

Reinforcement learning (RL) is an area of ML in which the agent takes action based on the variation of the environment to maximum long-term gains.^{66} The training process is aiming at finding a balance between exploration (of uncharted territory) and exploitation (of current knowledge).^{66} From 2014 to 2017, the presence of AlphaGo,^{67} a RL-based AI was able to beat top-notch Go players, showing the power of RL and its potential applications to materials problems such as interactive materials design.

Graph neural networks (GNNs), unlike standard neural networks operating on Euclidean data, operate on graphs that have non-Euclidean data structures consisting of nodes connected by edges without natural orders.^{68} Recent breakthroughs in GNNs, such as graph convolutional networks (GCNs),^{69} have demonstrated the capability of GNNs to learn graph embeddings through message passing between the nodes and its outstanding performance on semi-supervised classification tasks, which are potentially applicable to many materials and mechanics problems that inherently consist of graph structures.

Popular ML models and algorithms used in the design of mechanical materials, along with example applications, are tabulated in Table 1.

ML method | Characteristics | Example applications in mechanical materials design |
---|---|---|

Linear regression; polynomial regression | Model the linear or polynomial relationship between input and output variables | Modulus^{112} or strength^{123} prediction |

Support vector machine; SVR | Separate high-dimensional data space with one or a set of hyperplanes | Strength^{123} or hardness^{125} prediction; structural topology optimization^{159} |

Random forest | Construct multiple decision trees for classification or prediction | Modulus^{112} or toughness^{130} prediction |

Feedforward neural network (FFNN); MLP | Connect nodes (neurons) with information flowing in one direction | Prediction of modulus,^{97,112} strength,^{93} toughness^{130} or hardness;^{97} prediction of hyperelastic or plastic behaviors;^{143,145} identification of collision load conditions;^{147} design of spinodoid metamaterials^{163} |

CNNs | Capture features at different hierarchical levels by calculating convolutions; operate on pixel-based or voxel-based data | Prediction of strain fields^{104,105} or elastic properties^{102,103} of high-contrast composites, modulus of unidirectional composites,^{136} stress fields in cantilevered structures,^{137} or yield strength of additive-manufactured metals;^{121} prediction of fatigue crack propagation in polycrystalline alloys;^{140} prediction of crystal plasticity;^{120} design of tessellate composites;^{107–109} design of stretchable graphene kirigami;^{155} structural topology optimization^{156–158} |

Recurrent neural network (RNN); LSTM; GRU | Connect nodes (neurons) forming a directed graph with history information stored in hidden states; operate on sequential data | Prediction of fracture patterns in crystalline solids;^{114} prediction of plastic behaviors in heterogeneous materials;^{142,144} multi-scale modeling of porous media^{173} |

Generative adversarial networks (GANs) | Train two opponent neural networks to generate and discriminate separately until the two networks reach equilibrium; generate new data according to the distribution of training set | Prediction of modulus distribution by solving inverse elasticity problems;^{138} prediction of strain or stress fields in composites;^{139} composite design;^{164} structural topology optimization;^{165–167} architected materials design^{115} |

Gaussian process regression (GPR); Bayesian learning | Treat parameters as random variables and calculate the probability distribution of these variables; quantify the uncertainty of model predictions | Modulus^{122} or strength^{123,124} prediction; design of supercompressible and recoverable metamaterials^{110} |

Active learning | Interacts with a user on the fly for labeling new data; augment training data with post-hoc experiments or simulations | Strength prediction^{124} |

Genetic or evolutionary algorithms | Mimic evolutionary rules for optimizing objective function | Hardness prediction;^{126} designs of active materials;^{160,161} design of modular metamaterials^{162} |

Reinforcement learning | Maximize cumulative awards with agents reacting to the environments. | Deriving microstructure-based traction-separation laws^{174} |

Graph neural networks (GNNs) | Operate on non-Euclidean data structures; applicable tasks include link prediction, node classification and graph classification | Hardness prediction;^{127} architected materials design^{168} |

Database name | Material categories | Mechanical features | URL |
---|---|---|---|

AFLOW^{83} |
Alloys; inorganic compounds | Elastic properties | http://www.aflowlib.org/ |

Materials Project (MP)^{84} |
Inorganic compounds; nanoporous materials | Elastic properties | https://materialsproject.org/ |

MATDAT^{85} |
Steels; aluminum and titanium alloys; weld metals; etc. | Static properties; nonlinear stress-strain behaviors; cyclic stress–strain behaviors; fatigue behaviors | https://www.matdat.com |

MatWeb^{86} |
Polymers; metals; ceramics; semiconductors; fibers; etc. | Elastic properties; strength; toughness; hardness; etc. | http://www.matweb.com |

MatMatch^{87} |
Metals; composites; ceramics; polymers; glasses; etc. | Elastic properties; strength; toughness; hardness; etc. | https://matmatch.com |

MakeItForm^{88} |
Metals; polymers; ceramics | Elastic properties; strength; toughness; hardness; etc. | https://www.makeitfrom.com |

NIMS materials database (MatNavi)^{89} |
Polymers; inorganic materials; metals | Elastic properties; strength; hardness; etc. | https://mits.nims.go.jp/en/ |

Labeled datasets can be obtained from surveying the literature as well, such as datasets of copper alloys with different tensile strengths and electrical conductivities,^{93} ABO_{3} compounds,^{94} high-temperature ferroelectric perovskites,^{95} and single-molecule magnets.^{96} In addition, a glass dataset of experimental data was collected from both literature and existing databases.^{97} The size of the collected dataset relies heavily on the amount of accumulated literatures in the corresponding field. Relatively small datasets with tens to hundreds data points are acceptable for optimization approaches if equipped with an active learning loop.^{95,98}

Furthermore, text processing techniques can be utilized to replace manual labor in the extraction of features from research articles. With NLP techniques adopted, an automated workflow of article retrieval, text extraction and database construction was developed. to build a dataset of synthesis parameters across 30 different oxide systems, which is autonomously compiled and tabulated by training the text processing approach using over 640000 materials synthesis journal articles.^{99} The materials synthesis databases obtained from this approach enable a broader applications of ML methods than before, such as the prediction of materials synthesis conditions^{100} and candidate precursors for target materials.^{101}

Computational methods can be used to simulate materials of interest and relate the mechanical properties to the representative structures of the materials at different scales, from continuum to atomistic levels. For example, finite element method (FEM) was implemented to generate datasets of three-dimensional (3-D) microstructures of high-contrast composites,^{102–106} two-dimensional (2-D) tessellate composites,^{107–109} and metamaterials.^{110} Yang et al. created a dataset of synthetic microstructure images of materials with various compositional and dispersive patterns using Gaussian random field (GRF) method.^{111} High-throughput molecular dynamics (MD) can be utilized as a design space sampling method for the atomistic structures and behaviors of materials like silicate glasses,^{112} metal–organic frameworks (MOFs),^{113} as well as brittle materials with different crystal orientations.^{114}

A framework for data-driven analysis of materials has been built to avoid unacceptable computational expense of data generation from high-fidelity analyses, such as FEM simulations involving plasticity and damage, and reduced order methods were utilized to generate large databases suitable for ML.^{77,78,81} It is also possible to reduce the scale of design spaces by considering the symmetries in the materials problems to be investigated. The design spaces of 2-D tessellate composites under symmetric loadings can be truncated by half,^{107–109} and the generated topologies of architected materials were classified into 17 datasets according to the crystallographic symmetry groups in 2-D space.^{115}

Benchmark databases, such as MNIST,^{116} are particularly useful for comparing the accuracy and efficiency of various ML techniques on specific tasks. Recently, a benchmark dataset named Mechanical MNIST was constructed by converting the MNIST bitmap images into heterogeneous blocks of materials.^{117} This dataset, labeled by different forms of mechanical responses calculated from FEM simulations, can be used to evaluate the performance of metamodels of heterogeneous materials under large deformation.

Performing experiments to create sufficient large datasets for training DL models is currently difficult due to the extremely high cost. However, high-throughput experiments are applicable to the validation of trained ML models,^{118} and relatively small training sets can be augmented via post-hoc experiments in an active learning loop.^{95,98} Recently, an autonomous research system has been built to enable not only automated experimentation but also the selection of subsequent experiments under a framework of Bayesian optimization, which can be utilized to mechanical materials design problems such as optimization of additive manufacturing structures.^{119}

In a recent work on the prediction of fracture patterns in brittle materials, the discrete atoms in a triangular lattice, which is adopted from the MD simulations to generate the datasets of crack patterns, were mapped into ordered pixels in an image that can not only be treated as input to the first convolutional layer of the applied LSTM model but also eliminate the irrelevant information in the atomic structure other than the spatial features of the crack.^{114} In another example, least angle regression (LARS) was utilized as a feature selection algorithm for a large glass dataset taken from the literature and online databases.^{97} Image processing techniques, such as rescaling and cropping, were utilized to augment the initial dataset that might be insufficiently large to train a DL model.^{120} It has been demonstrated that less efforts on preprocessing are required to design features for DL than conventional ML methods due to the ability of the DL models to parse the representation from simple to abstract features through the training process.^{121} The techniques used to develop data-driven solvers might also inspire efficient methods to process sparse and noisy data of materials responses.^{70,71}

Materials with complex and disordered microstructures, such as glasses and alloys, typically have large databases obtained from experiments or simulations focusing on composition-property relationships. Thus, the selected features like concentrations of components are usually arranged as feature vectors, and ML methods good at processing input vectors are particularly suitable for the property prediction tasks of these materials. For instance, different ML algorithms (PR, LASSO, RF and MLP) were adopted to predict the Young's modulus of silicate glasses.^{112} Among those methods, MLP gives the highest accuracy, and the LASSO algorithm offers a slightly lower accuracy but higher simplicity and interpretability of the model. It is subsequently shown that using GPR instead of neural networks can avoid overfitting for a sparse dataset.^{122} Recently, a large dataset obtained from the literature and glass datasets was preprocessed to train deep FFNNs that allow the design of eight essential properties of oxide glasses, including Young's modulus, shear modulus and hardness.^{97} Wang et al. developed a design system based on neural networks for copper alloys that can rapidly screen the composition design space and provide the compositional design of new copper alloys with a target ultimate tensile strength and electrical conductivity.^{93} To discover strong and conductive copper alloys, Zhao et al. recently reported a systematic study of the selection of ML models (LIR, SVR, regression tree and GPR), dimensionality reduction techniques (principal component analysis, correlation-based and genetic algorithm) and additional features.^{123} For gradient nanostructured metals, Gaussian process based active learning surrogate models were developed to study the structural gradient effects on strength and deformation mechanisms.^{124} Furthermore, new superhard materials were proposed with the aid of ML techniques such as SVR,^{125} evolutionary algorithms,^{126} and GNNs.^{127} In a study by Wen et al., high entropy alloys predicted by the applied ML models were synthesized, showing higher hardness values than any other sample in the training dataset.^{128} ML models can also be trained to capture the relationship between salient structural features and mechanical properties. For example, deep neural networks that were trained to learn the relationship between the geometric patterns and mechanical responses of non-uniform cellular materials are capable of solving both forward and inverse problems.^{129} Liu et al. have achieved the fracture toughness prediction of polycrystalline silicon specimens using two different ML algorithms, RFs and FFNNs.^{130} In a recent study, the strength and toughness of spider webs were predicted by using a neural network trained with fiber lengths and orientations, as well as web connectivity and density.^{131}

ML-based prediction of mechanical properties can also be achieved using atomistic descriptors. For example, local properties (bond length, angle and dihedrals), global properties (density or ring sizes distribution) and porosity-related properties were fed as entries into a gradient boosting regressor to predict mechanical properties of zeolite frameworks.^{132,133} Given the system temperature, strain rate, vacancy defect and chirality, mechanical properties of single-layer graphene were predicted using different ML algorithms (stochastic gradient descent, k-nearest neighbors, SVR, DT, ANN).^{134} In a separate work by Moghadam et al., the relationship between the structure and mechanical stability of thousands of MOF materials has been established to predict the bulk modulus of MOF materials using an ANN that inputs structural or topological descriptors.^{113}

For materials that can be represented as tessellated spatial grids of multi-phase voxels, CNNs are advantageous over conventional ML methods in learning embeddings at different length scales ranging from voxels to representative volume elements (RVEs). The elastic deformation fields and effective elastic properties of high-contrast two-phase composites were predicted using 3-D CNN and datasets of 3-D volume elements with different microstructures (Fig. 3).^{102–105} Convolutional networks with different architectures were used to predict the mechanical properties of polymer nanocomposites based on microstructure images,^{135} thermo and mechanical properties of unidirectional composites,^{136} and stress fields in cantilevered structures.^{137} In particular, Herriott and Spear implemented two conventional ML methods (Ridge regression and gradient boosting) and a CNN model to predict the effective yield strength of additive-manufactured metals.^{121} When 3D images of the microstructures represented by crystal orientation are input to the CNN model, it outperforms the other two methods fed with microstructural features, demonstrating the strengths of CNN in learning higher-level features directly from image data and reducing the efforts on preprocessing and feature extraction.

Fig. 3 Predicting elastic behaviors of high-contrast composites using convolutional neural network (CNN). (a) An example microscale volume element, and (b) a comparison of strain field prediction from FEM and statistical models. (Licensed under CC-BY).^{104} (c) The compositional structures (top) and spatial statistics (bottom) of three example generated microstructure volume elements, (d) a schematic of the applied 3-D CNN architecture, and (e) a selection of three learned filters that help to distinguish microstructures similar to the three examples shown in (c), respectively (Reproduced with permission.^{103} Copyright 2017 Elsevier). |

The capability of generative models to deal with image-to-image translation tasks can be harnessed to achieve fast conversion between material distribution and mechanical fields. Ni and Gao developed a cGAN model to address the inverse elasticity problem of calculating elastic modulus distribution from observed displacement or strain fields in inclusion systems, mimicking an application scenario for real-time elastography and high-throughput non-destructive evaluation techniques.^{138} Recently, Yang et al. introduced a deep learning approach which predicts complex strain or stress fields of hierarchical composites directly from geometric information.^{139} Image-to-image translation using GANs has been implemented to investigate mechanical systems and exhibited astonishing performances in reproducing mechanical fields, extracting secondary information and extending to various loading conditions, component shapes and hierarchies. This framework could be further applicable to fast prediction of other physical fields with geometric information in image-based representation.

Mechanical problems involving nonlinearities such as plasticity, fracture and dynamic impact are known to be difficult and computationally expensive for conventional numerical simulation schemes. ML-based approaches have created new opportunities for addressing these long-standing problems.

For fracture problems, Pierson et al. developed a CNN-based methodology to predict the microstructure-sensitive propagation of a 3-D fatigue crack in a polycrystalline alloy based on the past crack surface.^{140} Guilleminot and Dolbow reported a data-driven framework that can generate new crack patterns in random heterogeneous microstructures through the combination of a manifold learning approach and a crack path reconstruction procedure.^{141} Moreover, Hsu et al. presented a ML-based approach combining convolutional layers and LSTM for predicting fracture patterns in crystalline solids based on atomistic molecular simulations (Fig. 4a).^{114} The proposed approach not only captures complex fracture processes but also shows good agreement regarding fracture toughness and crack length (Fig. 4b). The work further examined the crack propagation in more complicated crystal structures including bicrystalline materials and graded microstructures (Fig. 4c). The strong predictive power of their approach can be potentially applied to design materials with enhanced crack resistance.

Fig. 4 Predicting dynamical fracture using a deep learning approach, dependent on microstructural details. (a) Workflow of fracture patterns prediction. (b) Comparison of crack path, length and energy release between molecular simulations and the ML approach. (c) Prediction of crack patterns in bicrystalline and gradient materials (Reproduced with permission.^{114} Copyright 2020 Elsevier). |

For nonlinear deformation problems, Mozaffar et al. recently established a data-driven framework consisting of RNNs to learn history-dependent behaviors of heterogeneous RVEs loaded along different deformation paths, and it has enabled the prediction of plasticity-constitutive laws in an efficient and accurate manner without adopting the widely-used assumptions in existing plasticity theories (Fig. 5).^{142} Huang et al. developed a hyperelastic model using FFNNs and a plasticity framework via a combination of FFNNs and Proper Orthogonal Decomposition (POD).^{143} Yang et al. trained a deep residual network that can predict crystal plasticity using high-throughput discrete dislocation simulations.^{120} Wu et al. designed a RNN based on GRU to predict the stress–strain evolutions of elasto-plastic composite RVEs subjected to random loading paths.^{144} Yang et al. utilized ANNs to construct constitutive laws for isotropic hardening elastoplastic materials with complex microstructures.^{145} In a study by Zhou et al., a discrete dislocation dynamics model of straight dislocations on two parallel slip planes was self-consistently transformed into a continuum model via the integration of asymptotic analysis and ML methods.^{146} Chen et al. utilized DL models to find the inverse solution to collision load conditions with the post-collision plastic deformation of shell structures given.^{147} Stern et al. reported a framework for supervised learning in thin creased sheets which can not only accurately classify the patterns of training forces but also generalize to unseen test force patterns, demonstrating how learning can be achieved from plasticity and nonlinearities in materials.^{148} In order to solve both forward and inverse indentation problems, many efforts have been made using neural networks.^{149–153} Recently, Lu et al. demonstrated a general framework for extracting elastoplastic properties of materials from instrumented indentation results with significantly elevated accuracy and training efficiency, which have been furtherly improved by considering known physical and scaling laws and by utilizing transfer learning techniques when additional new experimental data are available.^{154}

Fig. 5 Learning history-dependent plasticity using recurrent neural networks. (a) Schematic of sampling temporally deformation paths. (b) A deformed heterogeneous representative volume element (RVE) with distributed circular fillers in the generated database. (c) Comparison of the results predicted by recurrent neural networks and calculated from FEM analyses for two different RVEs under different loading conditions (Licensed under CC-BY).^{142} |

2-D structures of materials can be represented as pixel images, fed as input to image processing models like CNNs and GANs. These models can significantly enlarge the design spaces to be explored for the optimal design, and the design process can be furtherly accelerated through the integration of appropriate optimization algorithms in the workflow. For instance, Gu et al. used CNN to design tessellate composites with optimized strength and fracture toughness (Fig. 6a–d).^{107,108} CNN was applied to extract local patterns of the composite around the crack tip in the framework. In these problems, the scale of the design space increases exponentially with the number of grid elements in the composites, and finding the optimal design can be easily intractable for brute-force approaches by elevating the grid resolution. In order to address this issue, Yu et al. integrated the CNN model with a genetic algorithm to accelerate the search process using the ML prediction as the fitness function for the optimization algorithm (Fig. 6e and f).^{109} In a study by Hanakata et al., a CNN-based search algorithm was developed to find optimal arrangements of kirigami cuts in graphenes to maximize stretchability.^{155}

Fig. 6 ML-based tessellate composites design for optimal strength and fracture toughness. (a) Workflow of the ML approach for the prediction of mechanical properties of composites. (b) Ranking comparison between the results from the ML approach and FEM simulations. (c) Optimal designs regarding strength and toughness in mode I test at various resolutions (Reproduced with permission.^{107} Copyright 2017 Elsevier). (d) Extended implementation to composites consisting of anisotropic building blocks (Licensed under CC-BY).^{108} (e) Framework embedded with genetic algorithm to accelerate the design process and (f) optimal designs in mode II test validated by MD simulations (Reproduced with permission.^{109} Copyright 2019 IOP Publishing Ltd). |

Encoder and decoder frameworks based on convolutional layers can be employed to accelerate the process of topology optimization of mechanical structures.^{156–158} Since the models were trained with the structures that have already been optimized by standard optimization methods, direct evaluation of mechanical properties (e.g., compliance) in loss functions can be avoided. As a trade-off, designs predicted by ML models may have mechanical incompatibility such as structural discontinuity, but these issues can be refined by connecting a cGAN model to the trained encoder and decoder network.^{156} Different from a pixel-based representation, a structural topology optimization method has been achieved through the movement of morphable components as basic building blocks, and both SVR and the k-nearest neighbors algorithm were adopted to extract the mapping between the external load and design parameters.^{159} Even though this approach shrinks the design space, it can avoid mesh dependency and model complexity issues induced by preprocessing structures into pixel images.

Topological design approaches using other ML techniques have also been widely reported in the literature. For example, structural designs of active composite beams and hard-magnetic soft active materials with target deflected shapes were obtained using evolutionary algorithms.^{160,161} Recently, Wu et al. reported an approach to design modular metamaterials using genetic algorithm and neural networks.^{162} They applied the method to the design problems of phononic metamaterials and optimization problems of interconnect for stretchable electronics. Kumar et al. built an inverse design framework of spinodoid metamaterials using deep neural networks that can provide optimal topologies for desired properties.^{163}

Leveraging the strengths of advanced ML techniques usually offers new pathways for the design of mechanical materials. Bayesian machine learning is a powerful approach for handling noisy data and can quantify the uncertainty of model predictions, which are particularly useful for design of metamaterials that are often sensitive to manufacturing imperfections. Bessa et al. demonstrated that data-driven designs of supercompressible and recoverable metamaterials made of brittle polymeric base materials can be found with the aid of Bayesian machine learning methods (Fig. 7).^{110} Generative methods have the ability to create plenty of new designs with different structures and even better mechanical performance compared to those in the training set, suitable for not only composite design,^{164} but also topology optimization.^{165–167} Mao et al. harnessed GANs to acquire hundreds of designs of 2D periodic units in architected materials that approach the Hashin-Shtrikman upper bounds and at the same time attain desired crystallographic symmetries and porosities.^{115} Other work reported the development of a semi-supervised approach to design architected materials using GNNs and the analogy between architected materials and graphs, that is, truss elements to edges, and truss pin joints to nodes.^{168} Graph connectivity and the load levels of a small fraction of nodes are fed as input to the GNNs that can predict the distribution of the load levels of the remaining nodes, and then the GNN model is integrated with a design algorithm to engineer the topological structures of the architected materials.

Fig. 7 Data-driven design of supercompressible and recoverable metamaterials using Bayesian machine learning. (a) Workflow of the data-driven design approach of supercompressible metamaterials. (b and c) Mechanical testing of the obtained designs of (b) a recoverable and highly compressible metamaterial produced by fused filament fabrication using polylactic acid, and (c) a monolithic metamaterial manufactured by two-photon nanolithography (scale bars, 50 μm) (Licensed under CC-BY).^{110} |

For instance, in order to solve nonlinear heterogeneous structure problems, neural networks have been used in a decoupled computational homogenization method where the effective strain-energy density is first computed at discrete points in a macroscopic strain space and then interpolated on RVEs.^{76} Inspired by the previous method, a data-driven framework aiming to model and design new composite material systems and structures has been built, accompanied with a method called self-consistent clustering analysis that make the framework applicable to materials problems involving irreversible deformation.^{78} Moreover, Liu et al. reported a data-driven method called deep material network, which is developed for structure–property predictions of heterogeneous materials under the effects of nonlinear, failure and interfacial behaviors.^{169–172}

Wang and Sun leveraged RNNs and the concept of directed graph to address the issues on the linkages between multi-scale models of porous media using a recursive data-driven approach, where the databases generated from smaller-scale simulations are used to train RNN models at larger scales (Fig. 8).^{173} They also implemented reinforcement learning to generate traction–separation laws for materials with heterogeneous microstructures.^{174} Capuano and Rimoli developed a new type of finite elements called “smart elements” in which ML models provide force predictions based on the elements’ states, circumventing the computation of internal displacement field and the need for numerical iterations.^{175} Chan et al. reported an unsupervised approach that combines techniques such as topology classification, image processing, and clustering algorithms to promptly identify and characterize microstructures, including grains in polycrystalline solids, voids in porous materials, and micellar distribution in complex solutions (Fig. 9).^{176} In a recent work by Samaniego et al., deep neural networks based on the variational form of the boundary value problems were implemented as solvers for partial differential equations (PDEs) in various solid mechanics problems, using a fundamental idea that the energy of the system to be minimized can be naturally treated as a loss function for the neural networks.^{177}

Fig. 8 A multi-scale multi-physics framework for poromechanics problems driven by directed graph representation and recurrent neural networks (Reproduced with permission.^{173} Copyright 2018 Elsevier). |

Fig. 9 An unsupervised approach for the identification and characterization of microstructures in 3-D samples of various material systems. (a) A workflow for autonomous microstructural characterization of 3-D polycrystalline solids. (b and c) Results of the ML-based microstructural analysis method on the analysis of (b) voids in porous materials and (c) micellar distribution in complex solutions (Licensed under CC-BY).^{176} |

ML approaches that can discover new physics may have a broad application in materials and mechanics researches. It has shown that ML can be trained to learn symbolic expression of physical laws. Well-known physics concepts including Hamiltonian, Lagrangian are predicted by symbolic regression.^{179} Brunton et al. revealed governing equations underlying a dynamical system with ML algorithms.^{180} Recent ML work using GNN has shown that the algorithms are capable to discover new analytical solutions for dark matter mass distribution.^{181} These works derived governing equations in a unique way and may offer a potential new direction for understanding the mechanisms and mechanical behaviors of various materials.

As summarized in this review, most of current researches focus on applying ML algorithms to solve materials and mechanics problems. Yet, it is worth pointing out that mechanical insights also have the potential to facilitate the development of ML. Geiger et al. showed that loss landscape of deep neural networks can be interpreted with a paradigm based on jamming transition.^{182} Inspired by information process in natural neural networks, spike neural networks (SNNs) transmit sparse and asynchronous binary signals between neurons which incorporates time into deep learning networks. As a consequence, SNNs have exhibited favorable properties including low power consumption, fast inference, and event-driven information processing.^{183} Despite the popularity of ML systems, they are arguably treated as “black boxes” due to the difficulty of inspecting how and why those algorithms can make accomplishments. The known knowledge in mechanics and materials science may help us understand the mechanisms behind ML algorithms and develop new learning techniques that can tackle challenging problems in materials design, such as design of hierarchical structures or multifunctional materials with desired overall performance of a set of material properties.

So far, the potential of using ML in design of mechanical materials has not been fully exploited yet with opportunities and challenges lying ahead to be explored and overcome. It is promising that ML-based approaches will revolutionize the way we understand and design materials.

ABBREVIATION | MEANING |

2-D | Two-dimensional |

3-D | Three-dimensional |

AI | Artificial intelligence |

ANN | Artificial neural network |

BL | Bayesian learning |

BP | Back propagation |

cGAN | conditional generative adversarial network |

CNN | Convolutional neural network |

DL | Deep learning |

DT | Decision tree |

FEM | Finite element method |

FFNN | Feedforward neural network |

GAN | Generative adversarial network |

GCN | Graph convolutional network |

GD | Gradient descent |

GNN | Graph neural network |

GPR | Gaussian process regression |

GRF | Gaussian random field |

GRU | Gated recurrent unit |

LASSO | Least absolute shrinkage and selection operator |

LIR | Linear regression |

LOR | Logistic regression |

LSTM | Long short-term memory |

MD | Molecular dynamics |

ML | Machine learning |

MLP | Multilayer perceptron |

MOF | Metal-organic framework |

MP | Materials Project |

NLP | Natural language processing |

PDE | Partial differential equation |

PINN | Physics-informed neural network |

POD | Proper Orthogonal Decomposition |

PR | Polynomial regression |

RF | Random forest |

RL | Reinforcement learning |

RNN | Recurrent neural network |

RVE | Representative volume element |

SNN | Spike neural network |

SVR | Support vector regression |

VAE | Variational autoencoder |

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