affil0affil0affiliationtext: Mechanics of Functional Materials Division,
Institute of Materials Science, Technische Universität Darmstadt, Darmstadt
64287, Germany**affiliationtext: Corresponding authors: xianglong.peng@tu-
darmstadt.de (Xiang-Long Peng<EMAIL_ADDRESS>(Bai-Xiang Xu)

# What can machine learning help with microstructure-informed materials
modeling and design?

Xiang-Long Peng  Mozhdeh Fathidoost  Binbin Lin  Yangyiwei Yang  Bai-Xiang Xu

###### Abstract

Machine learning techniques have been widely employed as effective tools in
addressing various engineering challenges in recent years, particularly for
the challenging task of microstructure-informed materials modeling. This work
provides a comprehensive review of the current machine learning-assisted and
data-driven advancements in this field, including microstructure
characterization and reconstruction, multiscale simulation, correlations among
process, microstructure, and properties, as well as microstructure
optimization and inverse design. It outlines the achievements of existing
research through best practices and suggests potential avenues for future
investigations. Moreover, it prepares the readers with educative instructions
of basic knowledge and an overview on machine learning, microstructure
descriptors and machine learning-assisted material modeling, lowering the
interdisciplinary hurdles. It should help to stimulate and attract more
research attention to the rapidly growing field of machine learning-based
modeling and design of microstructured materials.

Keywords— machine learning, microstructures, multiscale simulation, inverse
design, optimization

##  1 Introduction

Microstructured materials manifest in diverse engineering scenarios in forms
of polycrystalline microstructures, inclusion-matrix composites, bicontinuous
composites, porous structures [1], etc. Their macroscopic properties strongly
depend on the underlying microstructural features. This presents an avenue for
achieving tailored macroscopic properties which are unattainable in the base
materials. Thus, comprehending the quantitative impact of microstructures on
macroscopic properties is desired. The fabrication or synthesis of
microstructured materials is a complex process subjected to various process
parameters. These parameters significantly affect the resultant
microstructures, and thereby the corresponding macroscopic properties. Thus,
the design, fabrication, and application of microstructured materials
necessitate a deep understanding of the interplay between process and
microstructure, microstructure and property, and, ultimately, direct process-
property relations. Traditional experimental and numerical methods, though
valuable, cannot efficiently tackle these challenging tasks.

In recent years, artificial intelligence and machine learning (ML) have
emerged as transformative forces, driving profound economic and social
changes. They have become pivotal technologies in various research domains,
including material science and engineering [2]. The integration of ML and
data-driven techniques into scientific research methodologies has given rise
to what is known as the fourth research paradigm [3], extending the third
paradigm of computational science. This paradigm shift signifies a departure
from traditional scientific approaches, embracing the power of data-driven
insights and predictive capabilities facilitated by ML. A number of review
papers have illuminated the remarkable progress achieved through the
application of ML methods in different scenarios in the broad field of
material sciences and mechanics. These include materials design [4, 5, 6, 7,
8], atomistic simulations [2, 9], multiscale modeling and simulation [10],
mechanics of materials [11, 12, 13], etc. These reviews collectively
demonstrate the efficiency and effectiveness of ML methods in addressing a
spectrum of challenging problems in material design, modeling, and
engineering.

As summarized in Table 1, ML can assist materials modeling in terms of
constitutive surrogates, energy functional surrogates or field predictors, in
alignment with continuum mechanics theory. However, the existing research and
literature articles are mostly dedicated to homogeneous material or
homogenized microstructure. In fact, for these cases, traditional continuum
models have reached a high level of sophistication. The ML-based material
models frequently emerge only as alternatives or supplements to these
established frameworks. In contrast, there are hardly continuum models which
take microstructure descriptors or local properties as direct variables,
mainly due to the fact that the intricate and diverse nature of
microstructures and the mostly unknown material physics implications of
microstructures exceed the capabilities of conventional methodologies. In this
sense, microstructure-informed materials modeling presents an ideal
opportunity for ML-based modeling approaches to flourish.

Table 1: Classification of ML-based material models. Abbreviations: ”temp.”,
”chem.”, ”pot.”, and ”thermody.” denote ”temperature”, ”chemical”,
”potential”, and ”thermodynamically”, respectively.

[b]  Homogeneous Material Heterogeneous Material ML-based Models Input X
Output Y Input X Output Y I. Constitutive surrogate e.g., [14, 15] Physical
Fields: e.g., strain $\bm{\varepsilon}$, $\bm{\varepsilon}^{\mathrm{v}}$,
$\bm{\varepsilon}^{\mathrm{p}}$, electromagnetic field $\mathbf{E}$,
$\mathbf{H}$ Driving Forces: e.g., temp. gradient $\nabla T$, chem. pot.
gradient $\nabla\mu$ Thermody. Conjugated Quantities: e.g., stress
$\bm{\sigma}$, electromagnetic inductions $\mathbf{D}$, $\mathbf{B}$ Kinetic
Response: e.g., heat flux $\mathbf{j}_{q}$, chem. species flux
$\mathbf{j}_{c}$ Physical Fields: e.g., strain $\bm{\varepsilon}$,
$\bm{\varepsilon}^{\mathrm{v}}$, $\bm{\varepsilon}^{\mathrm{p}}$,
electromagnetic field $\mathbf{E}$, $\mathbf{H}$ Driving Forces: e.g., temp.
gradient $\nabla T$, chem. pot. gradient $\nabla\mu$ + Microstructural
Descriptors Thermody. Conjugated Quantities: e.g., stress $\bm{\sigma}$,
electromagnetic inductions $\mathbf{D}$, $\mathbf{B}$ Kinetic Response: e.g.,
heat flux $\mathbf{j}_{q}$, chem. species flux $\mathbf{j}_{c}$ II. Energy
functional surrogate e.g., [16, 17] Tensor Invariant: e.g., strain invariant
$I_{\bm{\varepsilon}}$,… $I_{\bm{\varepsilon}^{\mathrm{v}}}$,…
$I_{\bm{\varepsilon}^{\mathrm{p}}}$ Energy/Thermody. Potentials: e.g.,
dissipated energy $U$, free energy density $f$, $g$ Tensor Invariant: e.g.,
strain invariant $I_{\bm{\varepsilon}}$,…
$I_{\bm{\varepsilon}^{\mathrm{v}}}$,… $I_{\bm{\varepsilon}^{\mathrm{p}}}$ +
Microstructural Descriptors Energy/Thermody. Potentials: e.g., dissipated
energy $U$, free energy density $f$, $g$ III. Field predictor e.g., [18, 4,
19] Coordinates $\mathbf{r}$ Solution Fields e.g., displacement
$\mathbf{u}(\mathbf{r})$, stress $\bm{\sigma}(\mathbf{r})$, temperature
$T(\mathbf{r})$, electromagnetic inductions $\mathbf{D}(\mathbf{r})$,
$\mathbf{B}(\mathbf{r})$ Coordinates $\mathbf{r}$ + Local material properties:
e.g., mechanical properties $C(\mathbf{r})$, thermal conductivity
$\kappa(\mathbf{r})$, permittivity $\xi(\mathbf{r})$ Solution Fields: e.g.,
displacement $\mathbf{u}(\mathbf{r})$, stress $\bm{\sigma}(\mathbf{r})$,
temperature $T(\mathbf{r})$, electromagnetic inductions
$\mathbf{D}(\mathbf{r})$, $\mathbf{B}(\mathbf{r})$

Figure 1: a) Schematic of different aspects and their connections relevant to
ML-based modeling and design of microstructured materials. b) Illustration of
the general workflow for ML.

To promote this, we present a comprehensive review collectively addressing the
roles that ML can play in different areas relevant to the modeling and design
of microstructured materials including multiscale simulation, microstructure
correlation and sensitivity analysis, and microstructure optimization and
inverse design, as outlined in Fig. 1a. To this end, we highlight the
representative works/best practices from the literature to demonstrate the
potential of ML methods in solving challenging problems in the relevant
fields. Based on the reviewed works and our research experiences and
knowledge, we suggest the remaining challenges and urgent fields to which ML
can significantly contribute, aiming at stimulating and guiding future
investigations. Additionally, this work is valuable for instructive and
educative purposes, empowering researchers with the knowledge and confidence
needed to harness the potential of ML methods for advancing the modeling and
design of materials. The paper is structured as follows.

Section 2 introduces fundamentals on microstructure reconstruction and
characterization and ML methods. Modeling and designing microstructured
materials necessitate reconstructing microstructures and characterizing their
features, which is already a well-developed field with significant progress.
We will illustrate the relevant key points. Besides, a compact yet informative
introduction on the workflow of ML methods including task definition, data
collection and preprocessing, model selection, training, performance
evaluation, and deployment is provided. These bases are included for educative
and instructive purposes and are also necessary for the readers to understand
the content of the current work and the reviewed literature.

Section 3 is dedicated to data-driven multiscale simulation. The multiscale
nature of microstructured materials necessitates a multiscale modeling and
simulation strategy to characterize their properties and behaviors, which is a
challenging task. On the one hand, characterizing and reconstructing
microstructures (see Section 2) are non-trivial tasks, especially for
sophistical microstructures where high dimensional and diverse features are
involved. On the other hand, in a multiscale framework where the scale-
separation is assumed, a scale-bridging scheme is crucial. At the macroscale,
the object is effectively modeled as a homogeneous material at least locally.
The corresponding macroscopic/effective material properties/laws are obtained
through microscale simulations on representative volume elements (RVEs).
Existing multiscale methods are generally classified as concurrent (e.g.,
$\text{FE}^{2}$ method) or sequential types. The concurrent method couples
microscale and macroscale on-the-fly, ensuring high accuracy but is
computationally expensive. The sequential framework employs a pre-constructed
surrogate model for macroscale simulations, offering computational efficiency
but relying on experiences and lacking reliability for complex problems. These
drawbacks and limitations of traditional multiscale methods can be remedied by
ML-based surrogate modeling. Although the ML method can be applied to
homogeneous materials, we focus on those for inhomogeneous microstructured
materials. How these ML-based material models are integrated into to both
sequential and concurrent multiscale schemes will be discussed.

Section 4 addresses how ML methods are employed to construct correlations
among processes, microstructures, and properties. These correlations are
crucial for tailoring microstructures to achieve specific macroscopic
properties. However, the corresponding regression tasks typically involve
highly complex and nonlinear relationships with diverse and high-dimensional
inputs and outputs, posing challenges for conventional methodologies. In fact,
due to their powerful fitting capability, ML models are well-suited for
complex regression tasks. To unravel how process parameters influence
microstructural features and, subsequently, how these features impact material
properties, vast amounts of data are naturally required. This dataset can
originate from experimental measurements or traditional and ML-aided numerical
simulations. Subsequently, it is adopted to train ML surrogate models which
act as efficient predictors. The description of a process or a microstructure
may involve numerous parameters. However, not all of these parameters are
equally important in the correlations under investigation. With the aid of the
ML surrogate models, the sensitivity analysis can be conducted to identify the
subset of parameters that significantly influence the correlations.

Section 5 presents ML-based microstructure optimization and inverse design.
They play a critical role in designing microstructured materials with on-
demand properties. Microstructure optimization is to find a microstructure
that results in extreme properties (e.g., minimal or maximum). The inverse
design focuses on predicting a microstructure with specified target properties
not necessarily to be the extreme values. Both tasks are extremely challenging
due to their inherent high dimensionality, strong nonlinearity, and high
computational cost. In this regard, ML methods if properly employed can
efficiently solve these challenging tasks. In the optimization case, the ML
method can be directly used as an optimization algorithm such as Bayesian
optimization. Alternatively, incorporating the ML-based surrogate model into
the traditional optimization methods as an efficient property predictor can
largely promote efficiency. In the inverse design case, ML models can act as
surrogate models directly correlating the effective properties with
microstructural features.

Section 6 outlines the concluding remarks and perspectives. The scope of the
current work is not only on reviewing existing literature but also aims at
providing comprehensive perspectives and outlooks on the unexplored problems
and potential opportunities for advancements in the field of ML-aided modeling
and design of microstructured materials. Hopefully, it will inspire and guide
future research endeavors in this fascinating field.

##  2 Fundamentals

For educative and instructive purposes, we introduce some fundamentals on
microstructure characterization and reconstruction and ML methods in this
section, which are also essential for the readers to understand the content of
the current work and the relevant literature.

###  2.1 Microstructure Characterization and Reconstruction

Figure 2: Microstructure characterization and reconstruction. a)
Characterization of a two-phase microstructure by three different types of
representations, i.e., physical descriptors such as the particle size and
aspect ratio, statistical descriptions based on the two-point correlation
function, and ML-based representation illustrated by the feature maps learned
through the convolutional layers of a CNN model and a few examples of the
filters. b) The workflow of the Python package MCRpy for microstructure
characterization and reconstruction [20]. Figure b) is adapted from [20]
(published under license CC BY 4.0
https://creativecommons.org/licenses/by/4.0.).

Microstructure characterization and reconstruction is one of the key steps in
the process of modeling and design of microstructured materials. On the one
hand, investigating the microstructure-property correlation requires
characterizing the simulated microstructures. On the other hand, the
optimization and inverse design of microstructures necessitate the
reconstruction of microstructures with predicted microstructural features. The
literature offers various ML methods to represent and characterize
microstructures that can be integrated into computational and ML models for
property prediction and materials design [21, 22]. Details go beyond the scope
of this review article. Instead, a short summary on various microstructure
descriptors is given below, according to a broad classification:

  * •

Physical descriptors: Employing physical descriptors aims to characterize a
microstructure by a limited set of features. Physical descriptors are either
deterministic or statistical and can be classified into geometric,
compositional/chemical, or dispersional types. Composition descriptors
identify different phases within a material and quantify their respective
proportions, e.g., volume fractions and porosity. Dispersion status
descriptors characterize the spatial arrangement of different phases and their
proximity to neighboring elements, such as the nearest neighbor distance.
Geometry descriptors focus on the geometrical details, such as the size
distribution, surface area, and aspect ratio of inclusions or fibers in
composites. These descriptors possess clear physical meaning and are directly
interpretable for users. However, representing microstructures with such a few
descriptors may result in the loss of microstructure information. For this
reason, a feature selection process based on, e.g. variable ranking approach
and exploratory factor analysis [21] is necessary.

  * •

Statistical function based descriptors: Microstructures can be characterized
by utilizing statistical correlation functions, commonly known as n-point
correlation functions [23] and their variants, e.g., lineal-path function,
two-point cluster correlation function and spectral density function [21, 24].
These statistical functions essentially quantify the spatial correlations
among different points in a microstructure. This type of representation is
suitable for reconstructing statistically equivalent microstructures [25, 26].
However, the lengthy and abstract statistical functions-based descriptors
cannot be directly used for microstructure-property correlation. To remedy
this issue, dimensionality reduction techniques such as principal component
analysis (PCA) and linear/nonlinear embedding methods can be employed.

  * •

ML-represented descriptors: ML methods can be directly used for microstructure
characterization. For instance, taking the digitalized microstructure images
as inputs, convolutional neural network (CNN) models can extract
microstructure features by convolutional layers with different filters, which
are then converted into a low-dimensional vector by pooling and flattening
layers. Once well-trained by a large microstructure data, CNN models are
capable of characterizing the features of a system of microstructures. These
extracted features can act as inputs for further supervised and/or
unsupervised ML tasks such as classification and regression. Another type is
the instance-based learning [27]. It first creates a database that contains
representative sample instances, then defines the feature space (e.g., the
statistical functions) and similarity metrics applicable to all the samples in
the database (e.g. hierarchical support vector machine classification model),
and finally conducts either a brute force or algorithmic search to find the
most similar instance in the database upon query. In this case, not only the
database should be large enough but also the feasible space must be well
covered. Since the size of the database increases exponentially, it is
essential to develop a rapid searching mechanism that can efficiently locate
the most similar instance in the database upon a query.

As an example, we illustrate how the above three methods can be exploited to
represent the microstructural features of a two-phase matrix-inclusion
composite in Fig. 2a. In this example, the two-phase microstructure was first
represented by physical descriptors such as particle size and aspect ratio or
statistical descriptions based on the two-point correlation function.
Alternatively, it was characterized by features extracted via the
convolutional layers in a CNN model where different filters are applied to
obtain different features.

The other important part is microstructure reconstruction, which is the
reverse process of microstructure characterization. It enables the generation
of 2D and 3D microstructures with prescribed microstructural features (i.e.,
those based on the above introduced three types of descriptors) [1, 28, 20].
Depending on the microstructure characteristics, microstructure generation
could be implemented by various algorithms [1, 20]. For instance, Seibert et
al. [20] introduced a powerful platform for microstructure characterization
and reconstruction, i.e., MCRpy. The workflow of MCRpy is depicted in Fig 2b.
It treats the microstructure reconstruction as a modular and adaptable
optimization problem. And it enables the consideration of diverse descriptors
for microstructure characterization, arbitrary loss functions, and various
optimizers for microstructure reconstruction. Recently, it is demonstrated
that ML models such as the generative adversarial network (GAN) models can
efficiently reconstruct complicated microstructures (e.g., [24, 29]). More
information on microstructure characterization and reconstruction is found in
a relevant comprehensive review paper [21].

###  2.2 Fundamentals of Machine Learning

Before introducing how ML methods can advance the design and modeling of
microstructured materials, it is necessary to understand its fundamentals and
how it operates in practice. ML is a structured process that can be outlined
by a general workflow regardless of tasks and model architectures. As
illustrated in Fig. 1b, the general workflow includes:

  1. I.

Task definition: A ML process begins with defining the task and the objective
function. Depending on the tasks and objectives, ML can be broadly categorized
into three learning paradigms: supervised learning, unsupervised learning, and
reinforcement learning. The different tasks can be broadly defined as
regression, classification, dimensionality reduction and recommendation, as
described in many standard ML lecture notes and books (e.g., [30]). While
other types of tasks are equally important, we focus on regression problems as
they are closely related to material modeling and design of microstructured
materials in this paper.

  2. II.

Data strategy plan: Data collection steps follow, and preprocessing is then
performed to handle missing values, outliers, and inconsistencies. Data
scaling is necessary if the inputs and outputs vary over magnitudes. The data
is usually split into training and test sets.

  3. III.

Model selection: A suitable ML model is selected or developed based on the
task and data. Models can be generally classified into two main types:
conventional models (shallow learning models) and deep learning models.
Conventional models further fall into parametric and non-parametric
categories. Parametric models have a defined number of parameters, while non-
parametric models experience an increase in the number of parameters with
expanded input data [31]. One of the most simple ML models is the linear
model. Although it is brief, the linear model already includes the main
features of a ML model. It serves as a mathematical model describing the
overall linear relationship between a set of inputs and the corresponding
outputs. In this context, model training involves determining the parameters
(i.e., the slope and intercept) in the linear model to best fit the given data
points. Essentially, the goal is to minimize the difference between the model
prediction and the reference outputs for the given inputs. Other typical
conventional ML models include Decision trees [32], Random Forest, Support
Vector Machines, K-Nearest Neighbors, Gradient Boosting models, and Gaussian
Processes (see e.g., [33]). Different types of neural network (NN) models are
typical deep learning models. The architecture of NN models usually consists
of one input layer connected to an output layer by a number of connected
hidden layers. Each layer possesses a number of neurons. Each neuron is
characterized by a weight, a bias, and possibly an activation function (e.g.,
sigmoid, tanh, and relu) to introduce non-linearity. The depth and width of
the layers are critical factors. Depth refers to the number of hidden layers,
while width relates to the number of neurons in each layer. The trade-off
between depth and width is crucial, as a large width can hinder the learning
capability of the model. Achieving an optimal balance between the two is key
for efficient and accurate prediction performance.

Due to their powerful fitting capability, NN models are widely used for data-
driven material modeling and design. Artificial neural network models (ANNs)
are commonly used when the inputs are explicitly defined, e.g.,
microstructures represented by well-defined geometrical parameters or
statistical features. Convolutional Neural Networks (CNNs), Recurrent Neural
Networks (RNNs), Generative Adversarial Networks (GANs), and Graph Neural
Networks (GNNs) are among other well-known types of neural networks. CNN
models are suitable for situations where microstructure images act as the
inputs. RNN models are capable of modeling history-dependent physical
problems, e.g., viscoelastic and plastic problems. GAN models can be used to
generate new microstructures with features similar to those in the training
dataset. GNN models are preferred when the microstructures can be converted to
a graph consisting of nodes connected by edges, such as polycrystalline
microstructure and lattice structures. Besides, deep language models have also
been exploited to address different problems relevant to materials design and
modeling [30, 34].

  4. IV.

Training: The fundamental function of the ML workflow is its ability to learn,
facilitated by a relatively straightforward recipe and the utilization of two
key ingredients: data and a model. Additionally, an objective function, such
as mean squared error or cross entropy, is employed. This function, also known
as an error, cost, or loss function, calculates the average loss across the
entire training dataset. In essence, training involves an iterative
optimization process aimed at enhancing the model’s learning capabilities. It
should be noted that training is more than a pure optimization in the
mathematical sense. In traditional optimization, the objective is to
minimize/maximize a specific function based on the available data. However, in
ML, the objective is to minimize the error or loss of unseen data, namely the
generalization error. Hence, ML involves specific generalization strategies,
which are intentionally designed to sacrifice training error for
generalization error. For instance, methods are employed to modify the cost
function, by introducing a penalty term (L1 or L2 type) into the loss function
[30]. To efficiently train a ML model with a satisfactory performance,
hyperparameters such as the learning rate and the batch size should be
properly chosen. Note the difference between hyperparameters and parameters.
Hyperparameters are prior choices for the specification of model architecture
(e.g., layer and neuron numbers of a NN model) or for defining training
procedures (e.g., learning rate and regularization-related ones), while the
parameters of a model are determined through training.

  5. V.

Performance evaluation: Once a ML model is trained, its performance is
evaluated using predefined evaluation metrics on the validation or test data
set. This performance evaluation helps to understand whether the model is
underfitting, overfitting, or performing optimally. It guides the
hyperparameters tuning to attain a model with satisfactory performance.

  6. VI.

Deployment: Once the model is ready, it undergoes packaging, versioning, and
integration into the production environment. Continuous monitoring of the
model’s performance with incoming data is essential, and necessary updates are
made based on new data or requirements.

In practice, the best type of ML models and the corresponding hyperparameters
to choose depend on the characteristics of the tasks to solve, the nature of
the data (e.g., amount and distribution), and the available computational
sources. A good practice is to test multiple models and select the one with
the best performance. Experienced practitioners usually have good sense of
choosing hyperparameters, selecting/comparing various models along with
flexible data strategies.

##  3 Data-driven Multiscale Simulation

The literature review underscores the substantial potential of ML to enhance
materials modeling, highlighting a critical gap in microstructure-informed
materials modeling. While conventional constitutive models for homogeneous
materials are notably advanced and closely aligned with experimental
observations, ML-based material models in the literature often serve as
alternatives or complements to these established models. However, the
development of microstructure-informed models for microstructured materials
remains a significant challenge, with few conventional models available. In
this scenario, the complexity and diversity inherent in microstructures, along
with their complicated material behavior, surpass the capabilities of
conventional methods but offer an ideal platform for ML-based modeling
approaches. As summarized in Table 1, for both the cases of homogeneous
materials and microstructured materials, the ML-based material model takes
specified physical variables (e.g., physical fields, driving forces, or tensor
invariant) as inputs and predicts the corresponding physical outputs (e.g.,
thermodynamically conjugated quantities, and energy/thermodynamical
potentials) or effective material properties. Here we focus on the latter case
of microstructured materials for which microstructural descriptors may also be
introduced as inputs. In the subsequent discussion, we provide a summary of
current research initiatives aimed at addressing how these ML-based material
models are integrated into multiscale frameworks for modeling and simulation
of microstructured materials. In general, existing works on data-driven
multiscale simulation are classified as sequential and concurrent types. In
the sequential case, the data-driven method is adopted to predict the
effective (macroscopic) material properties or construct surrogate effective
material models. In the concurrent case, the constructed data-driven surrogate
material model is further incorporated into the macroscale simulations. In the
following, we will introduce some typical examples for each case.

Figure 3: ML-aided sequential multiscale simulation. a) Predicting the
effective thermal conductivity of composite structures as a function of
microstructural parameters and base material properties [35]. b) CNN-based
modeling of microstructure-dependent effective stiffness tensor of 3D
heterogeneous microstructures [36]. c) Surrogate modeling of elasto-plastic
constitutive relations in microstructured materials [14]. Figure a) is adapted
from [35] (own work published under license CC BY-NC-ND 4.0
https://creativecommons.org/licenses/by-nc-nd/4.0/). Figure b) is adapted from
[36] (published under license CC BY 4.0
https://creativecommons.org/licenses/by/4.0.). Figure c) is adapted from [14]
with permission from Elsevier.

###  3.1 Sequential Scheme: Effective Property Prediction

ML methods have been widely applied to model the microstructure-dependent
effective properties in microstructured materials. In these cases, the form of
the effective material model is usually known in prior and the corresponding
unknowns are the effective material parameters. For instance, effective linear
elastic behavior is characterized by the effective Hooke’s law, and the
effective thermal conducting is described by the effective Fourier’s law. In
this regard, the ML model takes the microstructure features and/or base
material properties as inputs and predicts the corresponding effective
material properties.

The authors have contributed several works in this field. Fathidoost et al.
[35] developed a gradient boosting-based ML surrogate model to predict the
effective thermal conductivity of particulate composite structures. The
microstructure features including the aspect ratio and volume fraction of
particles and the base material thermal properties including the interface
thermal resistance and thermal conductivity of the base materials are
considered as the inputs (see Fig. 3a). In Peng and Xu [37], ANN models are
constructed to predict the effective linear elastic properties including the
effective Young’s and shear moduli and Poisson’s ratio of composite triangular
lattice structures, where the structural features including the strut
thickness, strut angle, and Young’s modulus ratio between the two base
materials.

In the above two examples, the microstructure is well-defined by a few
microstructural descriptors. For disordered microstructures, more advanced ML
models should be exploited. As shown in Fig. 3b, a CNN model is proposed to
predict the effective stiffness of 3D random heterogeneous multi-phase
materials, where the microstructure images act as the inputs. Peng and Xu [38]
considered the data-driven surrogate modeling of effective ionic conductivity
of polycrystalline oxide ceramics with varying microstructural features. An
efficient GNN model is exploited to capture the features of individual grains
and grain boundaries as well as their interactions. There are also other works
on applying ML method to identify the effective properties of microstructured
materials, e.g., effective elastic properties of multi-material metamaterials
[39, 40], effective Young’s modulus of metal foams [41], and effective
magnetostriction of polycrystalline materials [42].

These surrogate models can act as the efficient property predictor for the
optimization and inverse design of microstructured materials (see Section 5).
However, the methodology discussed here only applies to the cases where
apparent macroscopic material parameters are of concern.

###  3.2 Sequential Scheme: Effective Constitutive Surrogates

The effective material behavior of microstructured materials is usually
dramatically different from the underlying base materials and, hence can
hardly be characterized by already known material models. In this regard, ML
models can act as surrogate effective material models. As summarized in Table
1, in this scenario, the inputs of the ML model are physical variables such as
physical fields, driving forces or tensor invariant and/or microstructural
features, while the outputs are thermodynamically conjugated quantities or
energy/thermodynamical potentials. In Fig. 3c, one example [14] is shown to
illustrate the general workflow to construct a data-driven surrogate material
model. In this work, the effective plastic behavior of a composite is
considered. Numerous finite element (FE) simulations at the RVE level are
performed to collect the dataset consisting of strain sequences paired with
the corresponding stress sequences. Then, a RNN model with time-history
prediction capability is constructed and trained with the dataset. The trained
RNN model is essentially an incremental elastic-plastic constitutive model.
Fernández et al. [16] developed an ANN-based hyperelastic constitutive model
for cubic lattice structures undergoing large deformation where both the
structure parameters and the strain tensor act as the inputs and the output is
the elastic potential. This ML-based material model can capture the strongly
nonlinear deformation behavior of the lattice structures with varying
structure parameters. Other related works are found in the review paper on
data-driven multiscale modeling [10] and the references therein. So far, only
very limited works considered the microstructure dependence in the ML model.
One reason is that much more data points are required to train the ML model
with the microstructure descriptors as the additional inputs.

###  3.3 Surrogates-enabled Concurrent Scheme

Figure 4: ML-aided concurrent multiscale simulation. a) Two-scale damage
analysis in Li-ion batteries [15]. b) Finite strain hyperelastic behavior of
composites [17]. Figure a) is adapted from [15] with permission from Elsevier.
Figure b) is adapted from [17] (published under license CC BY 4.0
https://creativecommons.org/licenses/by/4.0.).

Here, the data-driven concurrent scheme refers to the cases where the trained
surrogate models are further incorporated into the simulations at the
macroscale. Two such examples are shown in Fig. 4. As displayed in Fig. 4a,
Asheri et al. [15] proposed a data-driven multiscale strategy to predict
battery cell performance to bypass the high computational cost encountered by
the conventional FE2 method. A large dataset is obtained from simulations at
the microscale for electrolyte-active material two-phase RVEs with varying
material and state variables. The influence of the interface damage between
the solid electrolyte and cathode active material which plays a critical role
in cell performance and degradation is considered in a coupled chemo-
mechanical model. Based on the dataset, an ANN-based surrogate model is
constructed and trained, which can efficiently predict microscopic chemo-
mechanical behavior. The surrogate model is further integrated into a multi-
field two-level framework to predict cell performance, focusing on the impact
of interface damage on capacity loss. The performance of this data-driven
multiscale strategy is compared with the results from the original multi-field
two-level simulations. The comparison demonstrates that the data-driven
multiscale approach shows promising results with high accuracy and efficiency,
highlighting the potential of data-driven multiscale simulations for lithium-
ion battery design.

Kalina et al. [17] (Fig. 4b) developed a data-driven multiscale approach based
on physics-constrained NN models and automated data mining. To showcase the
methodology, they consider the effective hyperelastic behavior of a fiber-
reinforced composite. The physics-constrained ML model takes a set of
deformation tensor invariants as inputs and outputs the corresponding free
energy density. Unlike most data-driven multiscale methods, the ML model will
be re-trained during the simulation process if newly unseen deformation states
are observed. This is more efficient than directly training the ML model with
all the possible deformation states expected to occur in the macroscopic
deformation. The multiscale loop includes model training, macroscopic
simulation, data analysis and data enrichment.

Mianroodi et. al. [43] considered the ML-aided multiscale modeling of
nanoporous materials. To this end, a CNN model with microstructure images as
inputs and the corresponding elastic properties as outputs is trained based on
data attained from molecular statics calculations. This CNN model is further
incorporated into upper-scale FE simulations as the constitutive law. This ML-
based multiscale simulation is much more efficient than the full atomistic
simulation.

More related works are found in the review paper on data-driven multiscale
modeling [10] and the references therein. Multiscale modeling of composites
with spatially varying microstructures deserves more effort. In this regard,
the microstructure descriptor should be considered as additional inputs in the
surrogate model.

In general, the data required to train the ML-based material models can be
collected from experimental tests or generated by numerical simulations with
specified microscopic physical models. On the one hand, generating data from
simulations is usually much cheaper and enables high accuracy by precisely
controlling the relevant conditions and parameters. On the other hand,
experimental data that does not rely on assumptions and physical models can
better reflect real-world material behavior, and hence the resultant trained
ML models can be generalized and deployed for real-world applications. Due to
the limited available experimental data, in most existing works, only
simulation data is employed to train ML models. Although it may not be
feasible to solely rely on experimental data, a combination of simulation and
experimental data along with appropriate data augmentation techniques is
necessary to attain generalizable ML models for real-world applications.

##  4 ML-enabled Microstructure Correlation and Sensitivity Analysis

In the process of fabrication or synthesis of microstructured materials,
various process parameters are involved. These parameters determine the
resulting microstructures. The latter further affects the macroscopic
properties of the produced materials. Thus, revealing the intricate relations
among processes, microstructures, and properties enables the design and
fabrication of microstructured materials with target properties [44, 45].
These relationships are usually strongly nonlinear with various inputs and
outputs, and hence, are challenging for traditional methods to tackle. In
contrast, ML models with powerful fitting capability are well-suited for these
tasks.

The ML methods can be applied to construct process-microstructure (PM),
microstructure-property (MP), process-microstructure-property (PMP), and
process-property (PP) correlations. In this paper, we focus on MP and PM
correlations. Although a large number of process/microstructure parameters are
involved in these correlations, they are not equally important. In this
regard, the trained ML surrogate models can be exploited for sensitivity
analysis, which enables the identification of most relevant parameters for the
corresponding correlations [46, 47]. In the following, we will address these
aspects by introducing typical examples.

###  4.1 Microstructure-Property Relation

Here we will introduce how ML methods enable surrogate modeling of MP relation
by typical examples. Cases with different types of microstructure
characterization are separately discussed (see Section 2.1).

Figure 5: ML-aided modeling of microstructure-property relation. a)
Microstructure-dependent mechanical properties of paper materials [47]. b)
CNN-based end-to-end prediction of the mechanical field in 2D microstructures
of dual-phase steel under tensile loading [19]. c) CNN-based prediction of
microscale elastic strain field of 3D voxel-based microstructure of a high-
contrast two-phase composite [4]. Figure a) is adapted from [47] (own work
published under license CC BY-NC-ND 4.0
https://creativecommons.org/licenses/by-nc-nd/4.0/). Figure b) is adapted from
[19] (own work published under license CC BY 4.0
https://creativecommons.org/licenses/by/4.0.). Figure c) is adapted from [4]
with permission from Elsevier.

####  4.1.1 Descriptor-based

In this scenario, a microstructure is characterized by a few physical
descriptors. Xu et al. [48] considered MP relations of two-phase polymer
composites. The microstructures are represented by the volume fraction of
inclusion, the average nearest cluster center distances, and the distribution
of equivalent radius and elongation ratio of the inclusion, which quantify the
composition, dispersion status, and geometry, respectively. A data-driven
surrogate model is constructed to correlate the viscoelastic properties of the
composites with these microstructural features.

As reviewed in [24], the descriptor-based approach is widely used to
characterize the Li-ion battery microstructures. In this scenario, various
microstructural features are involved. The composition category includes the
volume fraction of active materials, binders, additives, and pores. The
specific surface and interface areas, surface-to-volume ratio, connectivity,
and tortuosity belong to the dispersion/agglomeration category. Particle size
(volume, equivalent diameter, diameter/radius along different axes), type of
particle shape/morphology/profile (sphere, ellipsoid, cylindrical, etc.),
sphericity, particle orientation, surface roughness, and pore
size/radius/diameter are among the geometry category. The correlation between
the microstructural features and the electrochemical and mechanical properties
of Lithium-ion battery materials are captured by ML models.

One of the works from our group [47] employed the feature-based ML approach to
correlate the paper microstructures with their mechanical properties (Fig.
5a). To this end, numerous microstructures with varying features are generated
and analyzed. Several features are extracted. FE simulations are then
conducted to determine their mechanical properties including the failure
strain, effective stiffness, and maximal stress under tensile testing. The
resulting dataset consisting of mechanical properties paired with the
structural features is adopted to train a ML model. This ML model is exploited
to conduct the sensitivity analysis to evaluate the importance of different
features on each mechanical property.

####  4.1.2 Correlation-function-based

The n-point correlation functions are commonly adopted to statistically
characterize disordered heterogeneous materials. In [49], a data-driven
framework is proposed to identify n-point correlation functions for
characterizing a number of microstructured materials including Sandstone,
metal-ceramic composites, Pb-Sn alloys, concrete, and particle-reinforced
composites. It is shown that the n-point correlations can be computed by image
convolution. Thus, a CNN model is constructed. The microstructure
reconstruction is considered as an optimization problem, i.e., finding a
microstructure with correlations matching the target correlations. A searching
algorithm is introduced to select a concise subset of n-point correlations.
The optimized microstructure representations are directly used to compute
material properties such as bulk/shear modulus and thermal/electrical
conductivity by adopting the effective medium theory. In such a manner, MP
relations can be built with significantly less data in comparison with purely
data-driven methods.

In [50], 2-point correlation functions have been exploited to characterize
microstructures of SiC fiber-reinforced titanium matrix composites. To this
end, three correlation functions including two for matrix/fiber auto-
correlations and one for matrix-fiber cross-correlation are introduced. PCA is
further adopted for dimensionality reduction so as to attain reliable low-
dimensional descriptors. FE simulations are conducted to obtain the ultimate
tensile strength of the microstructures, which together with the
microstructural features and the base material properties comprise the
dataset. With the aid of this dataset, a support vector regression ML model is
constructed and trained for the relevant MP correlation. The ML model is
further validated by experimental data.

####  4.1.3 ML Representation-based

Instead of relying on statistical or physical microstructure features, the
microstructure image itself can serve as the input for establishing the MP
relation. In this regard, all the microstructure information is resolved. CNN
models are commonly adopted for image-based MP correlation in 2D and 3D
microstructured materials.

A 2D example is illustrated in Fig. 5b. In this work by Lin et al. [19], CNN-
based surrogate models are constructed to predict von Mises stress and
equivalent plastic strain fields in commonly used dual-phase steels for
automotive applications. The ML model operates in an end-to-end manner, taking
2D segmented phase images from actual experimental scanning electron
micrographs as inputs and providing the mechanical fields as outputs. The CNN
models are trained using the U-net NN structure with data obtained from
numerous elastoplastic FE simulations of various microstructure samples. The
model performs well even for microstructures unseen in the training dataset. A
3D example [4] is shown in Fig. 5. In this work, a CNN model is applied to
predict the elastic strain field in 3D high-contrast two-phase microstructure.
It is demonstrated that CNN-based ML method is a promising technique to
construct feature-engineering-free models for MP correlations in sophisticated
microstructured materials. These two works on adopting CNN model for
prediction of the mechanical field distribution belong to category III in
Table 1.

One drawback of CNN-based surrogate models for MP relation is the high
computational cost due to the high dimensionality of the image-based inputs.
This issue is more evident for 3D microstructures.

###  4.2 Process-Microstructure Relation

Figure 6: Data-driven modeling of process-microstructure relation. a) Data-
driven modeling of correlation between process parameters and melting pool
features [33]. b) Data-driven modeling of grain structures as functions of
process parameters [33]. Figures a) and b) are adapted from [33] with
permission from Elsevier.

Fabricating or synthesizing materials with sophisticated microstructures is
the final step before their practical applications. These tasks are highly
challenging by conventional methods, especially when it comes to
microstructures with controllable features. In recent years, various additive
manufacturing (AM) techniques are rapidly developing. These ground-breaking
technologies enable the fabrication of complex microstructured materials with
different types of base materials. An AM process is usually governed by a
number of process parameters, which strongly affect the characteristics of the
produced microstructures and thereby their properties. Thus, it is vital to
understand the intricate process-microstructure correlations relevant to AM.
In this regard, ML methods can play important roles. There are already a few
review papers [33, 51] which well summarize some recent progresses in this
field. A few representative examples are introduced here.

The first step before starting an AM process is selecting the process
parameters such as the feeding rate, scanning speed, laser power and laser
parameters for producing a part with target features [33]. The intermediate
quantities such as the pool properties, the bead properties, and the
temperature fields and history can be directly correlated to these process
parameters by data-driven surrogate modeling (see Fig. 6a) [52, 53, 54, 55].

Monitoring and controlling of manufacturing process is of great importance to
perform on-the-fly quality control of the produced parts. Imani et al. [56]
develop a deep NN model to identify incipient flaws from layer-wise spatial
images. The good performance of the model on real-time flaw detection is
validated by experiments. This presents an opportunity to identify initial
defects in AM processes before completion.

Process parameters can also be directly correlated to the final
microstructural features, such as grain structures [57] (Fig. 6b), porosity
[58], and geometrical distortion [59, 60].

In most existing works on ML-enabled microstructure correlation, data is
generated by different types of simulations. Incorporating experimental data
into the ML model training is highly desired to enhance their performance in
practical applications.

##  5 Microstructure Optimization and Inverse Design

Optimization and inverse design of microstructures are two different
conceptions often mixed up in the literature. The former is aimed at finding a
microstructure with extreme properties (i.e., minimum or maximum) under
certain constraints. However, the latter focuses on attaining a microstructure
with specified target properties which are not necessary to be extreme.

Both tasks pose significant challenges due to their inherent high
dimensionality, strong nonlinearity, and computational demands. However, ML
methods offer efficient solutions to these challenges. In this section, we
will illustrate by representative examples how to exploit ML methods to
realize microstructure optimization and inverse design.

###  5.1 Microstructure Optimization

Optimizing microstructures to achieve extreme effective properties is a common
demand in the design process. As illustrated in Fig. 7a, the optimization is
generally an iterative process including proposing initial designs and then
updating them until some specified convergence criteria are fulfilled.
Depending on the applied algorithms, optimization methods can be categorized
as gradient-based (i.e., local) or gradient-free (i.e., global) types. In both
cases, the time-consuming property predicting is involved in each iteration.
In ML-based methods, property predicting is achieved by a trained ML model,
which speeds up the optimization process. In the following, we review
different ML-based optimization methods.

Figure 7: Data-driven microstructure a) optimization and b) inverse design:
Workflows and typical examples ([61, 62]). The right part of Figure a) is
adapted from [61] with permission from Elsevier. The right part of Figure b)
is adapted from [62] (published under license CC BY 4.0
https://creativecommons.org/licenses/by/4.0.).

####  5.1.1 Gradient Based-local Method

The topology optimization method is a frequently used gradient-based
optimization method to find out an optimized structure by tuning the material
or microstructure distribution. As a grade-based method, at each iteration,
the sensitivity of the objective function with respect to the design
parameters must be evaluated, which is usually achieved by solving certain
boundary value problems in a finely meshed domain. This makes the method
computationally expensive. Furthermore, in the multiscale topology
optimization case, the situation is even worse. At each iteration, similar to
the multiscale problem discussed in Section 3, the effective properties of the
microscopic properties are required at each material point in the macroscopic
simulation. As already illustrated by some recent works, data-driven methods
can help to remedy these issues and thereby dramatically increase
computational efficiency. Senhora et al. [63] proposed an efficient ML-based
topology optimization framework, where a CNN-based surrogate model is used to
predict the sensitivity for a given intermediate design, which speeds up the
optimization process. Recently, the data-driven multiscale topology
optimization method has been successfully exploited to design optimized
macroscopic structures based on different types of microstructures such as
isotruss metamaterials [64], spinodoid metamaterials [65], and strut-based
lattice structures [33]. In these works, ML models are introduced to map the
microscopic design parameters to the effective properties at the macroscopic
material point. Furthermore, the differentiable ML models can efficiently
provide the sensitivity of the effective properties with respect to the design
parameters, which further reduces the computational cost.

####  5.1.2 Global Method: Bayesian Optimization

Bayesian optimization is a data-driven method to optimize a black-box
objective function with a limited number of function evaluations. It consists
of two main parts: a statistical surrogate model (e.g., Gaussian Process) for
the objective function and an acquisition function based on the former for
determining the next sampling point towards maximizing the objective function.
The statistical surrogate model initially trained by a few data points is re-
trained after adding each newly sampled data point. The acquisition function
results from the trade-off between the exploitation and exploration. After a
number of iterations, even with a few data points, Bayesian optimization can
already help to find out a design with improved properties. Thus this method
is suitable for the cases where only a limited number of data points are
available. The example shown in Fig. 7a [61] illustrates the general workflow
of Bayesian optimization. The optimization of the strength and permeability of
triply periodic minimal surface (TPMS) structures is considered. To this end,
the structural parameters and properties of a few initial TPMS structures are
evaluated by simulations to generate the initial dataset, which is
subsequently applied to train the ML surrogate model. Then, the recommended
design is proposed based on the acquisition function. The properties of the
new design are calculated by simulations. The updated dataset with the newly
generated data is used to re-train the ML model. The above steps are repeated
until the optimized structure is found.

Bayesian optimization has also been widely used to optimize properties of
other microstructured materials, e.g., the transport properties of electrode
materials [66], the plastic properties of dual-phase steel microstructures
[67], the mechanical properties of hexagonal honeycomb metamaterials [68], the
rotation thermal modulation contrast ratio of multilayer hyperbolic
metamaterials [69], and the transport properties of graphene thermoelectrics
[70].

Bayesian optimization also has some limitations, e.g., it becomes extremely
computationally expensive when dealing with problems with high-dimensional
design spaces.

####  5.1.3 Global Method: Genetic Algorithm

The genetic algorithm is a typical gradient-free optimization method. It
starts with an initial population consisting of a number of randomly generated
designs. Then, the initial population is subjected to selection, crossover and
mutation to generate the updated population. This process is repeated until
the optimized design is attained. During the whole process, the evaluation of
the properties of the new designs are continuously desired, which could be
time-consuming. In this regard, the ML surrogate model can serve as the
efficient property predictor once been trained. Recently, by integrating ML
method with the genetic algorithm, the optimization of different effective
properties of microstructured materials are realized, e.g., the buckling
resistance of truss lattice materials [71], the customized loading curves of
the shell-based mechanical metamaterials [72], the effective Poisson’s ratio
of perforated auxetic metamaterials [73], and the fracture toughness of
polycrystalline materials [74].

###  5.2 Microstructure Inverse Design

Instead of finding a microstructure design with extreme properties, the
inverse design focuses on predicting a design with specified target
properties. In general, the data-driven optimization method can also be
applied for inverse design purposes provided the objective function is
suitably selected. However, by the optimization-based inverse design method,
an iteration process is required for each inverse design task. A more
efficient alternative is to construct a surrogate model which takes the target
properties as inputs and predict the corresponding microstructure design as
outputs. However, due to the fact that multiple microstructures may correspond
to the same effective properties, the inverse design task is generally ill-
posed. This issue can be remedied by the data-driven inverse design method.
Liu et al. [75] proposed a tandem network model by combining a forward and an
inverse neural network models. The workflow of this data-driven inverse design
method is illustrated in Fig. 7b by a typical example [62] where the inverse
design of effective elasticity of spinodoid metamaterials is investigated. The
forward NN model acting as the property predictor is trained first. Then, the
inverse NN model is trained by minimizing the reconstruction loss (i.e., the
error between the properties of the designed structure and the target values)
instead of the prediction loss (i.e., the error between the predicted
microstructural features and those corresponding to the inputs), where the
reconstruction loss is evaluated by the already trained forward NN model. This
ML-based direct inverse design strategy has been successfully applied to
realize the inverse design of different materials such as the transmission
spectrum of nanophotonic structures [75], the deformed configuration of shape-
programmable 3D kirigami metamaterials [76], the effective elasticity of truss
metamaterials [77], the effective elasticity of composite triangular lattice
structures [37], and the auxeticitiy of curved beam metamaterials [78].

The introduced data-driven inverse design method based on ANN models usually
requires a large dataset to train models with sufficiently high accuracy.
Currently, in most cases, the dataset is generated by inexpensive numerical
simulations. This method may be less efficient when the available data is
limited.

In summary, data-driven optimization/inverse design methods can efficiently
and accurately solve the optimization/inverse design tasks for various
microstructured materials with respect to different properties. Each method
has its own advantages and limitations. The reviewed representative works for
different methods can provide useful instructions on how to select the most
suitable one for a specified problem.

##  6 Concluding Remarks and Perspectives

In this paper, the recent contributions on applying ML methods to advance the
modeling and design of microstructured materials are reviewed. It is
illustrated that the ML methods can play their important roles in various
aspects relevant to the microstructure-informed materials modeling including
microstructure characterization and reconstruction, multiscale simulation,
process and/or microstructure-property correlation, and microstructure
optimization and inverse design. Although significant progress has been
achieved in this field, there are still important aspects desiring further
exploration. Based on our knowledge and understanding, we present the
following perspectives:

  1. a)

In the data-driven multiscale simulation framework, most ML-based surrogate
models are only applicable for a specified microstructure. To construct a
surrogate model generally applicable to at least a class of microstructures,
microstructural features should be considered as inputs in the ML models. In
this regard, a much larger dataset should be generated by sampling both the
loading conditions and the microstructural features. To reduce the number of
required data, one could adopt the automated data-mining strategy as suggested
in [17].

  2. b)

In the context of process-microstructure-property correlation, most works
focus on data-driven surrogate modeling of MP correlations. Limited works
consider the PM correlation. The direct PP correlation is rarely explored.
Future endeavors can be devoted to bridging the gaps and exploring the direct
linkage between the manufacturing process and resulting material properties.

  3. c)

In the context of microstructure optimization and inverse design, most works
focus on microstructures that can be well-defined by a limited number of
features. Addressing the more intricate task of optimizing and inversely
designing sophisticated microstructures demands more effort. Accomplishing
this challenging task requires the incorporation of more advanced ML models. A
noteworthy recent endeavor in this direction is presented in [67], where a
generative ML model is combined with Bayesian optimization to achieve the
inverse design of polycrystalline dual-phase steels, providing valuable
insights for future exploration.

  4. d)

The extrapolation capability of ML models are usually quite limited, i.e.,
they only performs well within the range of training dataset. In this regards,
adopting transfer learning may remedy this issue to some extent. Transfer
learning is a ML technique where a model trained on one task is reused or
adapted for a different but related task. It leverages knowledge gained from
one domain to another, potentially saving time and resources in training new
models from scratch. By fine-tuning or retraining only a portion of the model
on a smaller dataset relevant to the new task, one can adapt the pre-trained
model to perform well on the new task. ML models trained for a certain
microstructure type may be adapted to similar microstructures by re-training
with a much smaller dataset.

  5. e)

In the majority of existing works, the ML models are trained by data from
simulations. Although it is time-consuming and expensive to collect
experimental data, it may be beneficial to at least using a hybrid dataset
[33] consisting of both simulation and experiment data.

  6. f)

Effective curation and sharing of both data and code are crucial for enabling
practitioners to replicate, compare, reuse, and advance existing ML models, as
well as to construct innovative models by leveraging available datasets.
Consequently, this paves the way for the rapid and positive development of the
field of ML-assisted modeling and design of microstructured materials.

We anticipate that ML methods will become standard tools for addressing
engineering challenges, like traditional numerical methods such as the FE
method. Hopefully, this paper will contribute to paving the way for this
trend.

## Acknowledgements

B.-X. Xu acknowledges the financial support of German Science Foundation (DFG)
in the framework of the Collaborative Research Centre Transregio 270 (CRC-TRR
270, project number 405553726, sub-projects A06, B07, Z-INF) and 361 (CRC-TRR
361, project number 492661287, sub-projects A05), the Research Training Groups
2561 (GRK 2561, project number 413956820, sub-project A4), the Priority
Program 2256 (SPP 2256, project number 441153493) and 2122 (SPP 2122, project
number 493889809). The authors also greatly appreciate the access to the
Lichtenberg II High-Performance Computer (HPC) and the technique supports from
the HHLR, Technische Universität Darmstadt. The computating time on the HPC is
granted by the NHR4CES Resource Allocation Board under the project
”special00007”.

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