A critical review on the application of machine learning in supporting auxetic metamaterial design

The progress of machine learning (ML) in the past years has opened up new opportunities to the design of auxetic metamaterials. However, successful implementation of ML algorithms remains challenging, particularly for complex problems such as domain performance prediction and inverse design. In this paper, we first reviewed classic auxetic designs and summarized their variants in different applications. The enormous variant design space leads to challenges using traditional design or topology optimization. Therefore, we also investigated how ML techniques can help address design challenges of auxetic metamaterials and when researchers should deploy them. The theories behind the techniques are explained, along with practical application examples from the analyzed literature. The advantages and limitations of different ML algorithms are discussed and trends in the field are highlighted. Finally, two practical problems of ML-aided design, design scales and data collection are discussed.


Introduction
Auxetic materials stand for materials with a negative Poisson's ratio (NPR), which will expand laterally relative to the stretching strain direction [1].Auxetic behavior is relatively rare in conventional materials, primarily because it involves an increase in volume.This behavior is in contrast to the typical behavior observed in natural materials, where volume tends to remain constant.Compared with conventional materials, auxetic materials have many unique advantages in fracture resistance, indentation resistance, synclastic behavior, variable permeability, and energy absorption, which give them enormous potential in the fields of medicine, aerospace, sensors, and actuators [2].
For example, in a gas turbine, many perforated surfaces exist in the combustion chamber, the turbine section, the bypass duct, and the exhaust nozzle.Perforated surfaces with S-shaped holes have been developed by Javid et al [3] as an improvement for stress reduction based on the principle of the auxetic rotating square structures [4].This modification is the result of long-term research, transitioning from technical concept to practical application, and can be classified into several steps: the conversion from rotating square structures to low porosity metallic sheets [5], design improvements for stress reduction [3,[6][7][8], investigation about thermomechanical fatigue performance [9], and multifunctionality validation in actual application scenarios [10] as shown in figure 1.Other auxetic concepts have also been adopted in different fields, a few of which are listed here.The re-entrant structures have been applied in armor [11] and dampers [12] due to their great energy absorption capability.Chiral structures have been applied in medical applications, like medical stents [13] due to their curved geometry which has no sharp corners.Rotating rigid structures have been applied in load-bearing applications like structural elements and packaging [14,15] as they are in general stiffer due to their lower porosity than other auxetic designs.Origami-based structures have been applied in aerospace applications like solar panels [16] and in areas where non-porosity is crucial, such as heat exchangers [17].These auxetic applications primarily stem from their deliberate geometric designs, which are complex, multi-step workflows including auxetic concept selection, parameter tuning, and detailed product design to fulfill specific functionalities.[4]; (b) low porosity metallic structures [5]; (c) metamaterial with orthogonal S-shaped holes [3]; (d) thermomechanical fatigue tests [9]; (e) application in an annular combustor with the perforated material [10].
Conventional design methods could be easily directed in incorrect directions in an 'exhaustive search' , which consumes considerable time and resources [18].In contrast, machine learning (ML) can address nonlinear problems due to its expressiveness and flexibility, and once well trained can be used to achieve generalization and prediction at a high speed.With regression ML, surrogate models could be developed to replace the costly evaluation process.For example, Zhang et al [19] have developed surrogate models to predict the behavior of auxetic sleeve robots, replacing non-linear dynamic simulations and saving the evaluation time from minutes to instant.Vyavahare et al [20] have replaced the fabrication and testing process, which takes hours for one round, of 3D printed auxetic beams with surrogate models trained on experimental data.As experimental data is accurate but expensive, Hu et al [21] have transferred their surrogate model from a simulated dataset to a real experimental dataset, which provides a solution for both data availability and accuracy.Furthermore, some models could even provide domain prediction such as the stress and strain fields, allowing the metrics of evaluation (like maximum stress and fatigue life) to be calculated afterward [19].This makes it possible for future design tasks to benefit from prior works, even when the evaluation metrics have changed.Meanwhile, with the unique one-to-many mapping capability, generative ML models also have great potential in on-demand design problems that have multiple corresponding design solutions.Zhang et al [22] have adopted this kind of model for the design of the perforated auxetic sheets with the desired Young's modulus and stress level, which has reduced the optimization iterations to one-tenth of what was required without generative ML models.Considering the advantages of ML, the application of ML in the design of auxetic metamaterials deserves extensive study.
This review study intends to reveal how ML techniques have been and can be utilized for the design of auxetic metamaterials.The current auxetic design principles and how ML can support this process are the key focus of this study and are illustrated in depth as guidance for researchers who are interested in auxetic metamaterial or structure design.The main contributions of this work are summarized below: • Existing auxetic designs are summarized, highlighting four types of classic auxetic mechanisms and three types of variant designs.• The trend of design methods in auxetic metamaterials is analyzed.Especially, the ML-aided design methods including regression models for performance prediction and generative models for inverse designs are reviewed.Some practical problems in design tasks such as how to apply ML in different design scales and how to collect datasets for ML are also discussed in detail.• The research gap is identified in three aspects including the robustness of ML models, interpretable design models and more advanced hybrid models.
To the best of our knowledge, our study has gathered all research articles related to ML applications for auxetic metamaterials from 2000 to 2023.165 articles from journals and conference proceedings were included in this study.These articles covered topics including classic auxetic designs, their variants, and design methods.This study is the first to summarize the role of ML in supporting the design of auxetic metamaterials.Compared with the work of Wang et al [2] where auxetic metamaterials are surveyed, our work emphasizes how ML can support the design process and their implementation.Compared with the work of Guo et al [23] where the applications of ML in mechanical materials are surveyed, our work focuses on the realized and potential benefits that ML brings specifically to auxetic metamaterials.Researchers will have a clear view of the implementation of ML-aided design of auxetic metamaterials after reading this paper.The remainder of this paper is organized as follows.Section 2 summarized existing classic auxetic designs, their key characteristics and variant designs.Section 3 discusses the trend of auxetic design methods from expert-based, optimization-based to ML-aided design methods.Section 4 presents the application of surrogate (regression) models in performance prediction in two scenarios, value and domain prediction.Section 5 presents the application of generative models in inverse design and the hybrid method using both generative and regression models.Section 6 discusses different methods for design in different scales and methods for data collection.Section 7 presents remarks on the analyzed topics and possible future directions.

Auxetic metamaterials
Poisson's ratio is a measure of the Poisson's effect, which stands for the deformation of a material in directions perpendicular to the specific direction of loading and is defined as the negative ratio of transverse to axial strain [24].The term 'auxetics' meaning 'that which tends to increase' , was introduced into the study of materials with NPR by Evans in 1991 [25].Compared with conventional materials, auxetic materials have many unique properties which make them show enormous potential in many fields [26].In this case, auxetic metamaterials, a subset of mechanical metamaterials, have been adopted as a new perspective in research works to investigate their auxetic behaviors.
Mechanical metamaterials are materials with unique structures at the unit-cell level leading to mechanical properties unattainable by their constituent materials [27].These properties arise not just from the constituent materials but also from their specific geometric designs.Mechanical metamaterials often involve micro or meso-scale repetitive structural designs integrated into bulk materials.In other words, they are not homogeneous at a microscopic level but can be considered as homogeneous when viewed at scales much larger than individual unit cells.Different from structure design which mainly focuses on external loads and boundary conditions, metamaterials focus on not only small-scale structure designs but also the overall properties.
To investigate auxetic metamaterials, the first step is to understand the mechanism of the auxetic behavior, which can be traced back to some classic auxetic designs.

Classic auxetic designs
The four principal categories of the oldest and most emblematic auxetic designs include: re-entrant structures, which are distinguished by polygons having negative angles (exceeding 180 • ) as illustrated in figure 2(1); chiral structures, composed of central cylinders interconnected by tangential ligaments, depicted in figure 2(2); rotating rigid structures, as shown in figure 2(3), featuring rigid shapes jointed by hinges; and origami-based structures demonstrated in figure 2(4), employing the art of paper folding [28].All of them use internal structural mechanisms that create a rotational or leveraging effect of unit cell struts to the surrounding cells causing counterintuitive expansion [29].However, these structures exhibit significant variations in their characteristics.In general, chiral structures have relatively low stiffness because of the rotation of the cylinders; rotating rigid structures could offer a relatively high young's modulus but have serious stress concentration; the re-entrant structures could achieve the balance between these arguments, but they are typically anisotropic; the origami-based structures have a unique feature of zero or extremely low porosity [30].

Variants of classic auxetic structure
In section 2.1, we discussed four types of common auxetic structures.However, in actual engineering applications, these structures normally can not be directly applied.Classic structures are derived either from existing topology (i.e.re-entrant hexagonal structures are based on hexagonal honeycombs) or designed directly based on the unit cell behavior such as chiral and rotating rigid topologies.Consequently, these classic structures could have serious deficiencies, like stress concentration [39], low manufacturability [26] and low stiffness.Therefore, based on the classic structures, researchers have proposed many variant designs that meet specific requirements.As listed in table 1, these reported designs can be categorized into three groups including structural element transformation, hybrid structures, and hierarchical structures.Figure 3 shows examples of the variant design for re-entrant structures including structural element transformation to remove sharp corners (figure 3(a)), a hybrid of hexagonal and double arrowhead re-entrant structures (figure 3(b)), and first-order hierarchical re-entrant hexagonal structures [40] (figure 3(c)).The characteristics of these variant groups will be discussed in the following paragraphs.For a clearer illustration, specific examples are provided, and their corresponding figures are available in figure 4.
Hybrid structures consist of various types of auxetic structures or different sizes of unit cells.By adjusting the arrangement or combination of different unit cells, programmable auxetic behavior can be achieved.Wilt et al [29] (figure 4(2-a)) and Zhang et al [48] combined re-entrant honeycombs with different re-entrant angles to control the behavior of robot arms.Yao et al [49] (figure 4(2-b)) applied gradient patterns of centrosymmetric perforations to control the auxetic behavior.Zhao et al [50] (figure 4(2-c)) combined various re-entrant structures, effectively modifying the mechanical properties of the composite structures.Hierarchical structures, a class of systems composed of structural elements with their own structure, often have significantly enhanced mechanical properties compared to their non-hierarchical counterparts [51].This concept has been applied to the research of auxetic structures and has shown its great potential, in areas like stress reduction and specific modulus (stiffness/density ratio) enhancement.Regarding stress reduction, Morvaridi et al [52] (figure 4(3-a)) implemented sets of cuts characterised by different scales and Hou et al [40] (figure 4(3-b)) investigated the mechanical behaviors on hierarchical re-entrant honeycomb.In terms of enhancing specific modulus, the work of Sun and Pugno [53] and Taylor et al [54] (figure 4(3-c)) revealed that replacing hexagonal honeycombs with auxetic honeycombs in hierarchical structures enhances the elastic modulus.Also, Rayneau-Kirkhope [55] and Meza et al [56] explored the lattice-based auxetic metamaterials where the solid raw material was replaced by lattices, showing promise for lightweight applications.
Unfortunately, due to the complexity and diversity of variant designs, the development of a classic auxetic concept into an engineering application usually takes a long time [27].Therefore, lots of research efforts have been dedicated to deploying computer-aided techniques to explore the design space of metamaterials [57].These techniques and common design methods of auxetic metamaterials are discussed in section 3.

Designs and optimization of auxetic metamaterials
It has been decades since Lakes [58] created the first artificial auxetic foams.During this period, various types of auxetic structures and metamaterials have been proposed, while related research about their applications in different areas was gradually on the rise as listed in table 1.With different applications, the desired properties can be varied, from pure mechanical requirements for energy absorption [50,59] to more complex coupled requirements such as thermomechanical properties for cooling systems [9,10] and acoustic properties for wave propagation [60,61].Consequently, advanced design methods are needed for these complex requirements.

Variant designs
Propose paper and application

Traditional design method
Natural cellular structures are the result of natural evolution across billions of years, which normally have advanced performance in their applications.Therefore, when designing new structures, some researchers would abstract the theoretical model based on their expert knowledge in this area (re-entrant honeycombs [62] and rotating rigid structures [33]), while others would draw inspiration from the biological system and propose bionic models with similar functions (cancellous bones [63] and skins [64]).

Topology optimization
It is hard to get the optimal design by relying solely on experiential knowledge or traversing through all potential design spaces.To solve this problem, researchers applied topology optimization to the design of metamaterials, which could generate the best design under certain constraints [65][66][67].This approach has been applied to chiral structures [68][69][70][71], rotating rigid structures [72,73], bi-material designs [74] and re-entrant structures [75].

Machine learning aided design approach
Existing topology optimization methods are commonly aimed at optimization instead of on-demand inverse design.With the development of modern computation technology, machine learning (ML) has been growing rapidly as a computational solution for engineering problems that are difficult to solve in conventional approaches [76].Because ML requires little prior knowledge of the relationship between dependent and independent variables compared with the traditional method [77,78], it has been extensively applied in materials and structures, which demonstrates obvious superiority in terms of time efficiency and prediction accuracy [79].
In general, ML can be applied in two aspects of design processes, including metamaterial performance prediction and inverse design with desirable performance.In the first scenario, regression algorithms are typically used to predict the performance of metamaterials [78].In the second scenario, generative algorithms are commonly used to design the metamaterials for desirable mechanical responses [78].

Performance prediction
The design of metamaterials has relied heavily upon a 'forward loop' paradigm [80], where the designer would evaluate the proposed designs and optimize them iteratively based on the feedback from the evaluation process.As common approaches for the evaluation such as finite element analysis and experiments are time-consuming and expensive, a trade-off between performance and time is a common limitation of this method.Therefore, ML regression models have been developed as surrogate models to predict the performance of designs and accelerate the evaluations [29,[81][82][83][84].These models can be divided into two categories based on the dimension of prediction: direct value prediction of evaluation metrics and domain prediction for complex or undecided metrics.

Value prediction
The evaluation of a metamaterial normally will provide some metrics for tracking and comparing the performance of the proposed design, such as Poisson's ratio, Young's modulus, shear modulus, and bulk modulus [27].These metrics are quantitative values obtained from simulations or experiments using predefined equations.They will be then fed into some optimization process to obtain the optimal design or be used to create some knowledge map for reference in future work like Pareto front [10] or properties map in the design space [22].As the dimension of predicted metrics is relatively low, regression ML techniques are commonly applied in this problem such as Gaussian Process Regression (GPR), Random Forest (RF), Gradient Boosting Regressor (GBR) and regression models based on neural networks, with design information as input.The input may have various formats like the re-entrant angle as a design parameter, design geometry images for non-parameterized designs, and matrices of design parameters for unit cell arrangement, as shown in figure 5.

Gaussian process regression
GPR is based on the concept of Gaussian processes [85] and then this method was introduced in the field of machine learning in 1998 for regression and classification tasks [86].In GPR, the goal is to model the relationship between a set of inputs and their corresponding output.Given a training dataset, GPR updates the prior distribution to obtain the posterior distribution and then provides the distribution of possible output values [87].Because the prediction is in the form of a probability distribution, it can not only provide the predicted value (mean) but also the uncertainty of the current prediction (variance).
Song et al [88] developed a GPR surrogate model to estimate the bending and torsion stiffnesses of a concentric tube with a pattern hole.During the model development, they iteratively updated the training dataset from 300 to 325 samples.This update was based on new data points recommended by an acquisition function, which reduced the distribution variance.The developed surrogate model supported their multi-objective Bayesian optimization which improved the mechanical performance.Similarly, Wang et al [12] employed the GPR method on 16 samples to establish the metamodel of auxetic jounce bumpers for the optimization of the maximum vertical acceleration of vehicles when traveling through a bump.They have also optimized a sandwich panel with a three-dimensional double-V Auxetic structure core for blast energy absorption using a similar approach [11] on 50 samples.
Although GPR can be very effective for small datasets (dozens or hundreds of samples), especially when capturing uncertainty is important, it has a high computational complexity and a large space complexity because of the matrix inversion operation during training [87].With an increase in the amount of training data, GPR will not be applicable to large datasets.

Random forest and gradient boosting regressor
RF algorithm was proposed by Breiman [89], which combines the predictions of multiple decision trees [90] to make robust and accurate predictions.Also aiming at regression and classification tasks, RF is more practical and efficient for large datasets.
Tajalsir et al [91] developed an RF surrogate model on a large dataset of 3523 samples, which takes six parameters related to impacting and their hierarchical auxetic structures as the input vector to predict the dynamic impact stress.Also trained on a large dataset of 6598 samples, an RF model in the work of Ben-Yelun et al [92] predicted the mechanical properties of their auxetic unit cell including Young's modulus and Poisson's ratio using six geometrical parameters.Apart from the scalability, RF also has better interpretability and robustness [93], which is critical when understanding the prediction and applying large datasets with outliers.
Similar to RF, GBR is also a combination of multiple weak regression models, where each subsequent model is trained to correct the errors made by the previous models [94].Gaillac et al [95] applied GBR to predict bulk and shear moduli of auxetic zeolites relying on geometric features.They claimed that GBR models are considered robust, interpretable, and applicable for their small dataset (121 pure silica zeolites [96]).

Neural networks based models
Instead of GPR, RF, or GBR, neural network-based models have become the most popular solutions in the domain of surrogate models due to their flexible architectures and capability to model complex relationships.Although the concept of artificial neural networks (ANNs) can track back to the work of McCulloch and Pitts [97], the dominant position of ANNs in the field of machine learning was established in the mid-2010s, driven by large datasets, improved computational resources, and the release of open-source frameworks like TensorFlow [98] and PyTorch [99].Considering that the application of ML is currently in a rapidly evolving stage, it is not as semantically rigid as other fields.Therefore, before we proceed, it is important to clarify several concepts.An ANN is a model with interconnected nodes, called neurons, organized into layers (input, hidden, output).Each neuron applies a nonlinear activation function to its input and passes the output to the next layer [100].Deep neural network (DNN) is an ANN with multiple hidden layers (depth) between the input and output layers, enabling it to learn complex representations of data through multiple levels of abstraction [93].Multi-layer perceptron (MLP) is a type of ANN with an input layer, one or more hidden layers, and an output layer, where each neuron applies a weighted sum of inputs, passes it through an activation function, and is fully connected to neurons in the subsequent layer [100].In short, MLPs are a specific type of ANN, whereas DNNs represent a broader class of ANNs with multiple hidden layers, and both MLPs and DNNs are instances of ANNs.However, as MLPs are the oldest form of neural networks, in literature, MLPs, ANNs, and DNNs are easily confused.Except MLP, there are other types of ANN that take high-dimension input to achieve value prediction like convolutional neural networks (CNNs).
• MLP, being fully connected networks, are typically applied when the input and output are both of relatively low dimension, like design parameters and evaluation metrics.Wang et al [101], Du et al [102], Wang et al [103], Zhang et al [104] and Bronder et al [105] predicted the Poisson's ratio of their auxetic metamaterials using MLPs with their design parameters or design genes (also vectors) as inputs.Pham et al [106] developed an MLP using six design parameters to predict the natural frequency of auxetic honeycomb sandwich plates.Bronder et al [105], Carakapurwa and Santosa [107] and Gao et al [108] predicted the energy absorption capacity of their designs using five, three, and two design parameters, respectively.In addition, Hammad and Mondal [109] developed an MLP-based auto-encoder trained on non-auxetic designs to detect these auxetic designs by reconstruction errors.They also developed an MLP to predict Poisson's ratio with the latent representation of designs.Grednev et al [110] implemented and compared a range of machine learning models, including MLP, k-nearest neighbors regression, support vector regression, and XGBoost, regarding the aptitude to predict stress-strain curves under quasi-static compressive loading where MLP is the most proficient model.The predicted metrics can then be used for optimizing the metamaterial design.Chang et al [111], Dong et al [81] and Liu et al [112] developed MLP models and then accelerated the following inverse design using genetic algorithm (GA) optimization for re-entrant structure, cross-chiral metamaterial and auxetic metamaterials with peanut-shaped pores.Other than GA, Vyavahare et al [20] applied the gray relational analysis technique to improve flexural responses and to reduce the weight and fabrication time of their auxetic metamaterials.They also demonstrated that a deeper MLP exhibits advantages in modeling material defects.
ANNs can not only accelerate optimization by predicting evaluation results, but most of them also provide an explicit function between the input and output.Because of the explicit formulation, Liao et al [82]  conducted a straightforward design sensitivity analysis during optimizing a tetrachiral auxetic to achieve the target Poisson's ratio with a restriction at a required stiffness.
To solve the low interpretability problem of ANNs, Lyngdoh et al [83] applied Shapley additive explanations to interpret their predicted results and found that the fraction of voids has the highest influence on the Poisson's ratio of cementitious cellular composites.
As the network architecture can be very flexible, MLPs can not only make low-dimension predictions from high-dimension inputs but also get high-dimension predictions from low-dimension inputs.Wu et al [113] developed an MLP using sixteen input signals from an auxetic-interlaced glove to recognize the full English alphabet (twenty-six outputs).However, although MLPs are adaptable between different input and output dimensions, for tasks related to images, they are capable of prediction [114], but they do not excel at it.Developing MLPs for images requires a massive number of neurons to handle such pixel-level input or output effectively, which leads to the problem of overfitting and makes training computationally expensive.
• CNN, which consists of convolution layers, activation functions, and pooling layers [115], is the most common network architecture for image-related tasks.In value prediction tasks related to auxetic metamaterials, images of design geometry can be used as the inputs of surrogate models.Unlike MLPs, convolutional layers in CNNs focus on neighboring pixels, and the same kernel is applied across different parts of the image in each layer.These features make CNNs able to recognize patterns regardless of their positions and much more effective in the aspect of model size compared with MLPs.For example, Zhang et al [116] used cameras positioned within the hollow interiors of handed shearing auxetic actuators to record deformation during motion and then maps the visual feedback to the actuator's tip pose.Wilt et al [29] developed a CNN using an 11 * 11 image (stands for an input auxetic lattice with 11 * 11 unit cells) to predict the calculated mean absolute error (MAE) of displacement between the predefined optimal design and input design.In both works, their CNNs encoded the input images into smaller latent matrices, which were then expanded as latent vectors and fed into subsequent MLPs for value prediction.

Domain prediction
However, in some scenarios, the evaluation metrics are not easily predicted or formulated.For example, the maximum von-Mise stress could happen in different locations in different designs [22], which might not be highly related to the design parameter.As a result, the relationship between the input and the output of surrogate models could be too abstract to capture.Meanwhile, surrogate models with specified output metrics cannot be carried over when the evaluation metrics change.Especially considering the training of ML models becomes more costly as model complexity increases, researchers are increasingly interested in foundation models that can be applied to different metrics.
In this case, if the model could predict the response of metamaterial in the target domain like the displacement or stress field in finite element analysis, it will be more convenient to obtain the evaluation metrics by extracting the corresponding data from the predicted domain.On the other hand, if the definitions of design parameters are modified, a similar mismatch problem could occur between the original model input and new parameters.The model will also be more universally applicable if it takes the non-processed designs as inputs like the images or geometry.As shown in figure 6, the ML model could still predict the stress field even if the geometry has been modified to reduce stress concentration.

CNN
As mentioned above, treating the input and output as images in 2D problems or 3D voxels in 3D problems seems like the most intuitive solution.For ML tasks related to images or voxels, such high dimensional tasks are mainly conducted by CNNs [117], as discussed in section 4.1.3.Unlike value prediction tasks where the input images will be first encoded by CNNs and then processed by some MLPs to obtain the values, in domain prediction, CNNs are constructed using an encoder-decoder formula, as the output domain is normally identical to the input design.Another advantage of CNNs is their abundant literature data from computer vision which provides experience in network architectures and model development.For example, Nie et al [118] developed the StressNet with Squeeze-Excitation blocks to predict the stress distribution in cantilever beams.Also, similar blocks have been adapted in a modified U-net [119] developed by Croom et al [120] to predict the stress around defects in additively manufactured metals.Both models provide stress prediction of every pixel, which is the foundation of further improvement for stress reduction.Zhang et al [19] reproduced their works for the displacement prediction of re-entrant auxetic metamaterials and both of them achieved good performance in the aspect of MAE.
There are also limitations for CNNs.In most CNN-related works, data for model development is collected from the discretized domain (mesh) of simulations, then interpolated into the grid format and fed into CNNs.This is because CNNs constrain input and output fields to be defined on rectangular domains represented by regular grids.This interpolation operation could introduce errors due to the mismatch of pixels and nodes in the mesh.

Graph neural networks (GNN)
To solve this mismatch, recently, GNNs [121] have been applied to predict physics in the real world.A graph is a mathematical structure that can model the pairwise relations between objects, which matches the concept of discretization in most simulations.From particle-based fluid simulations as highlighted in Belbute-Peres et al [122] to mesh-based simulations for aerodynamics, structural mechanics, and flag dynamics as detailed in Pfaff et al [123], GNNs have demonstrated remarkable capabilities.
Like convolution in CNN, in graphs, a parallel concept termed 'message-passing' (MP) has been employed to formulate GNNs, where the local node is computed as an aggregation of information from adjacent nodes.If k MP layers were connected together, the local node can accumulate information from k-hop neighbors, thereby expanding its receptive field and gaining an awareness overall graph structure.The overall awareness is necessary, as the behavior of a particular node might be significantly influenced by nodes that are relatively far away, such as those where boundary conditions are applied.
Yang and Buehler [124] utilized an eight-layer GNN model to predict the local atomic stress of 3D Graphene unit cells under certain mechanical tests using its initial atomic coordinates as the input.As their input is still at the unit-cell level, an eight-layer model is deep enough for their task.However, due to the need for fine meshes with lots of elements in complex designs, the 'over smoothing' effect of deep GNNs [125] on large graphs and limited GPU memory become inevitable problems.To solve this, Zhang et al [19] proposed a multi-scale GNN, which combined the U-net architecture with MeshPool, a multi-scale geometry-informed pooling technique.They also proposed distortion loss, a geometry-informed loss function inspired by element distortion in meshes, which provides an extra geometric constraint to the GNN.They applied their method on meshes with thousands of nodes and proved the advantage of GNNs in the aspect of gradient prediction compared with CNNs.This is vital for tasks like stress predictions, as the research interests are normally related to the stress concentration area instead of the MAE of stress.

Mesh-free NN
In both CNN and GNN surrogate models, designs are fed into the models in the form of high-dimensional data like images and graphs to achieve high-dimensional prediction.However, researchers have also developed models that provide domain prediction with low dimensional input and output.
Raissi et al [126] developed a physics-informed deep learning framework, which takes the coordinate of a target point and problem settings as the input to predict the response at the target point.This framework provided a unique mesh-free algorithm, as the differential operators in the governing PDEs are approximated by automatic differentiation [127].Although we have not seen any application in auxetic metamaterials, we believe it is a promising framework in domain prediction.

Inverse design
As we have discussed in section 2.2, more advanced and complex metamaterials have been developed for variant purposes.Consequently, the dimensions of the design space increased exponentially [80].In this case, there could be multiple designs that achieved similar performance.On the other hand, there is a strong possibility that the optimizer could be trapped at local minima points [128].To obtain all potential designs, optimization-based methods with ML surrogate models need multiple initial start points for the optimization and computational acceleration of surrogate models has rapidly diminishing returns as design complexity grows [80].Therefore, a fundamentally different approach called 'inverse design' is proposed.Inverse design is a very intuitive approach, in which the desired properties (such as Poisson's ratio and Young's modulus) are fed into some algorithms and the algorithms generate the corresponding designs.This method theoretically avoids some of the computational challenges associated with forward problems [80].
Similar to what we have discussed in section 4.1.3,researchers have developed ANNs for design prediction with performance requirements as input.Kollmann et al [129] have developed a CNN model that takes three input parameters to produce an output image with material density on every pixel.But they also mentioned that different designs would be generated if they added and modified an extra image to represent the initial design.This is caused by the one-to-many mapping relationship between the performance requirements and potential designs, which can not be learned by regression models.In machine learning, to solve the 'one-to-many mapping' problem, generative models are introduced.

Generative model
The concept of the generative model came from the area of statistical classification, proposed by Ng and Jordan [130], who categorized the statistical models into two types, generative classifiers, and regression classifiers.Generative classifiers provide the joint possibility distribution between the observation and the target instead of the conditional distribution of the target under the observation compared with regression classifiers [130].

MDN
The MDN was proposed by Bishop in 1994 [141].Unlike standard NNs update their weights based on the errors between the ground truth values and the output values, MDNs use the output discrete value from NNs to create a mixed Gaussian distribution and then, train the NNs to achieve consistency between the training dataset and the mixed distribution.Because of its high accuracy in low dimensional design space compared with other generative models according to the previous research about inverse architecture benchmark [142], MDNs are commonly chosen for parameter-based inverse design [134].In auxetic-related works, Zhang et al [22] have developed an MDN to generate design parameters for the perforated auxetic sheets.Their benchmark between the MDN and other generative models also demonstrated the advantage of their MDN in the aspect of design accuracy and diversity.
However, MDNs have two major limitations, including the limited design dimension and the predefined number of mixtures.As new designs are sampled from the mixed distribution, if there are n design parameters, the number of neurons in the output layer is 3 * m * n, where m is the number of mixture and 3 corresponds to three values (mean, variance and weight) for a sub-distribution [22].Therefore, MDNs are not commonly used for generating images that need a large number of pixels (high dimensional problems).Also, the number of mixtures constrained the maximum number of unique designs for one input requirement.These limitations make MDNs not applicable to complex design problems.

GAN
GANs are models which have a generator network and a discriminator network compete to improve each other's performance to generate data gradually.In the field of inverse design, as the generated designs need to satisfy the desired design requirement, conditional GANs (cGANs) [143], are more popular.In cGANs, generators take in random noise along with the condition to generate synthetic designs, while discriminators take in designs (either real from the dataset or synthetic from the generator) and the same condition to predict the probability that the input data is real.
CGANs can be constructed using either MLPs or CNNs, which provide the capability of generating low-dimensional design parameter sets or high-dimensional images.This feature makes it possible to create designs without a set of predefined parameters and expand the design space.
Zheng et al [144] developed a cGAN that takes Young's modulus and Poisson's ratio as input labels to generate 256 * 256 pixelated geometric patterns.They created a large dataset with 100 000 designs for their model development because of the challenging training process of cGANs.Although cGANs are relatively new and advanced generative models, the training of cGANs is often challenging due to the unstable adversarial training, lack of convergence criteria and the mode collapse phenomenon [145].

VAE
VAEs extract the features from the input data and represent the data as a small-dimensional latent vector, which follows a predefined distribution (normally Gaussian distribution) [138].New latent vectors will be sampled from this distribution and then fed into a decoder to recreate the input data.Conditional VAEs (cGANs) are also more popular in inverse tasks, which take in conditions and latent vectors to generate designs.
Similar to cGANs, cVAEs can also handle both low-dimensional (MLP-based) and high-dimensional (CNN-based) tasks.In addition, they are generally easier to train than cGANs because they have a more stable and well-defined objective function.However, samples generated by cVAEs are often blurrier compared to those generated by cGANs because of the averaging effect and regularization terms in VAEs [146].
Kim et al [147] generated 50 * 50 pixelated microstructures with require Poisson's ratios using a vector-quantized cVAE, which is an extension of the cVAE suitable for a discrete representation of the latent space [148].The vector-quantized cVAE is selected to solve the above-mentioned blur problem of cVAEs.They believe that the binary selection of specific pixels may discretize the response surface of physical properties and their model achieved much clearer designs than conventional cVAEs even with a small training dataset.

Hybrid inverse design
Instead of just using generative models, hybrid models of generative and regression (discriminative) models have also been investigated for data-driven design.Two kinds of hybrid models have been reviewed.In the first kind of hybrid model, a low-quality result from the regression model will be enhanced by the following generative model.For example, Yu et al [149] developed a CNN-cGAN hybrid model in which the cGAN is used to improve the resolution of the initial design prediction from the CNN.But in this case, the advantage of design diversity was constrained, as the initial design already defined the topology of the structure.Another kind of hybrid model uses generative models to predict the initial designs and then further optimizes these designs with regression models.A hybrid model of a GAN and a CNN was developed by Tan et al [150], in which the GAN was trained to capture the geometric constraints applied to the microstructures, and the surrogate CNN was trained to evaluate generated designs.An optimization process was conducted on the latent vector of the GAN to obtain microstructures with desired properties.This method achieved diversified generative design with different initialization of the latent vector.However, without labels as conditions, generated designs from the GAN might not be close to the optimal designs, resulting in more iterations in the optimization process.Similiar, Zheng et al [151] developed a VAE to abstract the truss designs into a 48-dimension reduced and continuous latent space, then optimize the latent vector to obtain desired designs.In the work of Zhang et al [22], an MDN is developed to provide both diversified and desired design candidates corresponding to input labels.These candidates were then optimized by an MLP to improve the design accuracy.
These works focus on diversity and accuracy.But the authors believe that there are still multiple aspects that can be considered in the hybrid models.For example, a sub-model for manufacturing difficulty could provide suggestions to the generative model to remove features that are difficult to manufacture.A similar combination could be implemented from many perspectives, including cost, environmental impact, or some human interaction components to reflect the preferences of designers.

Discussion
In this section, two problems in ML-aided auxetic metamaterial design are discussed including the design scale (unit cell design and metamaterial design) and the data collection.Table 2 summarized the reviewed papers in these aspects.Additionally, we also provided a general reference workflow of how to conduct ML-aided auxetic designs and summarized how researchers customize their ML model for the auxetic design tasks.

Design scales 6.1.1. Unit cell design
Unit cell design involves designing the basic repeating unit or module that forms the basis of larger metamaterials or systems.Most unit cells [82,83,105,112] in auxetic metamaterials are parameterized designs based on classic auxetic designs.Researchers took advantage of existing auxetic knowledge and designed their auxetic variants for their purposes.As the key elements (such as the re-entrant angles [31], the chiral rib number [153], and the rotating geometries [154]) of classic auxetic designs have been well investigated, design parameters related to these elements will be inherited in these variants.These parameterized designs are normally achieved by MLP-related design methods, as discussed in section 4.1.3.There are also unconventional unit cells that are difficult to be parameterized, such as the Voronoi-based auxetic unit cells proposed by Zheng et al [144].For these designs, high-dimensional methods like GAN and VAE will be very helpful.

Metamaterial design
Metamaterial design in mechanical engineering focuses on the arrangement and integration of unit cells to form a complete mechanical system.Depending on the level of complexity, metamaterials can be categorized into three categories including homogeneous metamaterials, periodic and non-periodic heterogeneous metamaterials, as shown in figure 7.
Homogeneous metamaterials (figure 7(a)) refer to metamaterials with uniform and consistent unit cells throughout the metamaterials.This uniformity allows for the assumption that the metamaterial is constant throughout its volume.Therefore, the design of homogeneous metamaterials can be simplified as the design of the unit cell with homogenization assumptions.For example, Liao et al [82] abstracted the minimum representative volume element and imposed a macro strain with periodic boundary conditions to simulate the behavior of tetrachiral metamaterial.
Unlike homogeneous metamaterials, heterogeneous metamaterials exhibit heterogeneity at various scales within the metamaterials.The simplest heterogeneous metamaterials are periodic or pseudo-periodic designs with variable porosity (figure 7(b)).Pseudo-periodic metamaterials consist of unit cells with the same topology but with distortions, such as conformal cellular structures [155], while unit cells of periodic metamaterials are all the same.For both periodic and pseudo-periodic designs, as the topology of unit cells is uniform and normally predefined, the design problem will be simplified as to determine the arrangement of design parameters throughout the metamaterials like porosity.If the arrangement is controlled by a function with a low dimensional tunable vector, it can be obtained by an optimization-based method.For example, Gao et al [108] adjusted the porosity of honeycombs in the y direction using the aspect ratio of cell wall thickness and length, which is controlled by the minimum and maximum ratio on the bottom and top of the specimen.Then they iteratively search for the best pair of rations to maximize the average energy absorption.
If the arrangement needs to be directly assigned by a two-dimensional (pixels of images) or even a three-dimensional (voxels of volume) matrix, the optimization-based method might be not applicable.Wilt et al [29] proposed an approach by predicting the performance difference between a predefined desired design image and 500 randomly sampled design images using CNN model to obtain the one with the minimum difference.Zhang et al [156] simulated the elastic modulus of the re-entrant lattice with different re-entrant angles then mapped four types of unit cells into the topology optimization result to obtain the maximum stiffness or the maximum stiffness with minimum Poisson's ratio.
Complex heterogeneous metamaterials (figure 7(c)) have design freedom in every unit cell (non-periodic).In this case, unit cells could have typologies and porosity that differ from each other, making the design task very challenging.In this case, the only auxetic-related work is conducted by Zeng et al [152], in which they developed a multi-network deep learning system for the design of gradient mechanical metamaterials.In their case study of the human hip joint, they predefined two images corresponding to Young's modulus and Poisson's ratio requirement of every unit cell then generated these unit cells using their generative model with a filter applied to guarantee the connectivity between unit cells.There are two major challenges to applying their method in other tasks.Firstly, a predefined map of unit cell properties is not common in design tasks.Instead, most tasks have loads or boundary conditions as well as design objectives.Secondly, in their work, the influence of neighboring unit cells on the properties of local unit cells is not considered.In other words, they used periodic boundary conditions to obtain the training data but their actual metamaterial is not homogeneous.

Data collection
ML-aided design is often data-driven because most models require a training process on a prepared dataset.Therefore, how to collect such datasets is a critical problem, especially when considering that the data availability and quality will directly affect the model selection and accuracy.For data availability, due to the cost of real-world experiments, almost all ML-aided design works reviewed in this paper used simulated data, from larger scales like whole system simulations (a vehicle virtual prototype [12]) and metamaterial application simulations (auxetic lattice tubes [29,48], energy absorption blocks [108]) to the unit cell scales (tetrachiral auxetics [82]).To validate the quality of simulated data, some work [19,81,105,111,112,156,157] did real-world experiments to test the consistency between real-world and simulated data.Only the work of Vyavahare et al [20] applied thirty-two real-world specimens to train their MLP due to the lack of a well-established simulation for the fused deposition modeling (FDM) process.They investigated the effect of design factors (angle, width, and length of arm) of FDM-manufactured auxetic unit cells on their mechanical responses.
Another aspect of data quality is data distribution.It should be considered that the accuracy of ML models is approximately positively correlated with data density.Hence, it is preferred to have more data points in the areas of the design space with desired performance.However, as the performance maps in the design space are normally unknown, the common methods of data sampling are uniform sampling [22] (figure 8(a)), random sampling Wilt et al [29] (figure 8(b)) and their hybrids like Latin hypercube sampling.Considering these methods can not enhance the interested area of design space, the concept of active learning [158] was introduced, in which ML models would iteratively update datasets with data points they want to learn from, as shown in figure 8(c).For example, Gao et al [108] developed their MLP from a dataset of 100 randomly selected designs then interactively adding the top 100 performers into the training dataset.They proved that their model will converge to the same level in the 6th generation even with different initial datasets.
There are certain scenarios in which ML models can be utilized without a traditional dataset, such as reinforcement learning and physics-informed ML.However, the development of evaluation environments for reinforcement learning and residual loss terms for physics-informed ML is very challenging and has not been seen in auxetic works.

ML-aided auxetic design workflow
Although ML is a remarkably powerful tool as discussed above, effectively utilizing these capabilities still presents significant challenges.This complexity primarily stems from the inherent flexibility and extensive customization options that ML models offer.Therefore, an ML-aided auxetic design workflow, including model selection, data preprocessing, hyper-parameter tuning, and other customization is formulated here.
The first step of ML-aided auxetic design is to determine what ML model to use, which is related to the purpose of using ML.As reviewed in this paper, there are two main tasks: performance prediction and inverse design.In the auxetic scenario, if the design concepts have been determined and the design has been parameterized, such as the re-entrant angle in re-entrant structures or perforation ratio in rotating rigid structures, normally these designs can be then represented as a low-dimensional vector.In this case, if the purpose is performance prediction, ML models like GPR, RF/GBR, or MLP can be applied for value prediction such as Poisson's ratio and Young's modules, while field prediction such as stress distribution can theoretically be accomplished using connected CNN architectures.However, in practice, it is very challenging to predict a high-dimensional field from a low-dimensional vector.For non-parameterized designs, according to the way of representing the design, CNN (for image) or GNN (for graph) is advisable to consider.If the goal is to generate designs based on required properties, generative models should be considered.For low-dimensional design parameter generation, MDN might have better accuracy but can not generate high-dimensional designs, while VAE and GAN can generate both.To further improve the inverse design performance, the generated designs can be evaluated and optimized using surrogate models.
Once the ML model is selected, the next step is to conduct data collection and data preprocessing.For data collection, if a specific data region is targeted, such as designs with lower stress concentration or reduced manufacturing cost, active learning sampling might be beneficial to increase data density.In the absence of such preferences, or when an initial dataset is needed for active learning, random or uniform sampling should be employed.For data preprocessing, not only standard operations like data cleaning of missing/error values and data normalization/standardization but also some auxetic-related characteristics should be considered.For example, in stress reduction tasks, where stress levels vary significantly, model performance in low-stress areas may be inferior to that in high-stress areas if training is based on mean or squared error [22].But low-stress areas are normally the desired area in stress reduction tasks.Given that percentage-related errors are less stable, employing a hybrid loss function might yield better results.For deformation-related tasks, predicting deformation across the entire design may not be necessary due to the continuity of material displacement.Thus, dimension reduction techniques, such as using continuous functions to describe deformation, should be considered [29].Additionally, aligning data with the network structure is crucial.For example, when using a CNN to process images of a unit cell with periodic boundary conditions, the input images must be also patterned periodically (instead of using zero padding) for accurate boundary predictions [48].
For hyper-parameter tuning, there are many parameters that can change the performance of ML models, such as the learning rate, the number of layers, the number of neurons in each layer, batch normalization, dropout, attention, and so on.To obtain the optimal or semi-optimal parameters, some hyper-parameter tuning software, like NNI [159] and Hyperopt [160], can be used.
Finally, there is also some customization possible in other aspects such as network architecture and customized loss terms.Regarding network architecture, specific customization is tailored for certain design representations.Zheng et al [151] regraded the lattice as graphs and created two MLPs to process the node feature and adjacent matrix separately, then merged them to obtain the latent representation for generation and prediction tasks.Zhang et al [19] proposed a pooling method on meshes to reduce graph size, enabling the development of a graph-based U-net.There is also network architecture customization for multi-module input like multi-scale [161] and time-sequenced inputs [19].Some customization, like the Squeeze-Excitation Residual Network [120] may not be directly related to the physics of auxetic designs but have indeed improved model performance.In the term of customized loss terms, there are works such as physics-informed loss terms related to potential energy [162], residual equations [163], and geometric constraints [19].However, it is worth noting that most of the reviewed works still utilize standard ML workflow, with common network structures adopted from the computer science domain.

Conclusions
This review paper aims to summarise the current auxetic design principles and how ML can support their design process to facilitate and enable the subsequent works of ML-aided design.
To provide sufficient background for auxetics to readers, four classic auxetic designs have been exhaustively discussed including re-entrant, chiral, rotating rigid, and origami-based structures, which are the foundation of various auxetic metamaterials in different domains.A new way to categorize these auxetic metamaterials based on their evolvement from the classic designs is presented and summarized in table 1.With respect to the challenge of design space introduced by the variant designs, the paper explored the evolution of auxetic design methodologies, transitioning from expert-driven and optimization-based approaches to those supported by machine learning.
In this paper, diverse ML algorithms are illustrated to understand how they can help tackle the challenges associated with auxetic designs.Table 2 summarizes the papers that utilized ML techniques for the design of auxetic metamaterials.In general, value prediction of performance prediction is the scenario where ML techniques are most frequently implemented.In value prediction, GPR models are used for regression tasks with requirements of confidence level on small datasets.RF models have good scalability on a large dataset, better interpretability, and robustness compared with GPR models.GBR is similar to RF but has better performance on small datasets.Finally, neural network-based models, as the most popular ML models, dominate the field of performance prediction because of their flexibility in architectures and capability of modeling complex relationships between any data type (design parameters or images).In addition to value prediction, for some scenarios where the evaluation metrics are not easy to be predicted or formulated, surrogate models for domain prediction have been proposed using CNNs or GNNs.Although CNN is the dominating architecture in computer vision, GNNs naturally accommodate mesh-based or particle-based analysis, which makes them perform better in the aspect of gradient and a promising architecture.For inverse designs, researchers have adapted generative models, from low-dimensional design parameters generation (MDN) to high-dimensional geometries or image generation (GAN, VAE).Researchers have also developed hybrid models of generative and regression models to achieve both diverse and accurate designs.Some practical problems of applying machine learning including design for different scales and dataset collection are also discussed.For design in unit cell scale, unit cells are either defined by low dimensional data like parameters or by high dimensional data like images or graphs.Low-dimensional data can be designed with the help of GPR, RF, GBR, MLP, and MDN, while high-dimensional data can be obtained by CNN, GNN, GAN, and VAE.In comparison, design at the metamaterial level is more challenging, especially for heterogeneous metamaterial which still have limitations to applying ML in the design process.As ML is driven by data, it is vital to have a good training dataset.Most works are using simulated datasets and some of them conducted real-world experiments to validate their simulation.To improve the data distribution, the concepts of active learning have also been introduced to the design sampling of simulated datasets.But for topics without well-established simulation, real-world data have also been applied with carefully designed ML models to avoid the over-fitting problem due to the limited quantity.There are also data-free ML algorithms, like reinforcement learning and Physics-informed ML, but they have not been applied in auxetic design yet.
With the development of ML-aided auxetic design summarized, the future research trends can be estimated: • Most performance-predicting models only rely on training data, which makes their predictions physically irrational due to over-fitting.To increase the robustness, ML models could be costumed by introducing more physics knowledge in the aspects of architecture, loss function and so on, especially in models like GNNs where meshes and graphs are perfectly matching each other.• Existing works related to inverse design usually emphasize the accuracy or diversity of designs.The interpretability of the generated design is ignored, which is very important for human designers to learn from the design algorithm.Some techniques of interpretable ML should be adopted in auxetic design.• The hybrid inverse design is still in the early stages.Sub-models from different aspects such as manufacture difficulty, cost, environmental impact, and heterogeneous metamaterial design could be considered as part of the hybrid models.

Figure 5 .
Figure 5. Value prediction example: predict performance metrics of re-entrant structure using design parameters, design geometries or matrices of unit cell arrangement.

Figure 6 .
Figure 6.Domain prediction example: predict von Mises stress of the re-entrant structure using design geometries or design meshes.

Figure 7 .
Figure 7. Three types of metamaterial design (a) homogeneous metamaterials; (b) Periodic heterogeneous metamaterials with predefined topology; (c) Non-periodic heterogeneous metamaterials with design freedom in every unit cell.

Figure 8 .
Figure 8. Three types of data sampling techniques: uniform, random, and active learning sampling.

Table 1 .
Reviewed papers related to auextic variant designs.

Table 2 .
Reviewed papers related to auxetic metamaterials design using ML.