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

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


Introduction
Mechanochemical spinodal decomposition refers to a continuous phase transformation mechanism due to an onset of instability with respect to the composition and/or a structural order parameter. It occurs in materials systems with a free-energy density that is non-convex in strain-composition space. Wide regimes of the state space lie far from thermodynamic equilibrium, and the resulting first-order dynamics manifests in evolving microstructures that are distinguishable by strain and composition variables [1]. Mechanochemical spinodal decomposition exists in many important multi-component crystalline solids, such as cubic yttria-stabilized zirconia, lithium-ion battery electrode material Li x Mn 2 O 4 , transition metal hydrides and certain two-dimensional materials such as TaS. In such material systems, as the first-order dynamics is driven by fluxes determined by the local free energy density, the material microstructure, delineated by strain and composition variables, undergoes changes. The macroscopic behaviors and properties are inherently related to the evolving microstructures. Progress has been made in understanding the detailed dynamics and in modeling the resulting microstructures [1,2]. However, in order to optimize the properties of existing materials and to design new materials, it also is essential to rapidly predict the material's macroscopic response based on the detailed microstructure.
Macroscopic material responses/properties can be measured from well-designed experiments or predicted from physics-based direct numerical simulations (DNS). Numerical methods to upscale the nonlinear macroscopic behav-ior of a heterogeneous microstructure are commonly categorized as computational homogenization methods. They necessitate the solution of expensive boundary value problems (BVPs) on representative volume elements (RVEs) that encompass the targeted material microstructures [3,4]. It is impractical, if not impossible, to evaluate macroscopic material properties based on either experimental measurements or DNS when a large number of microstructures need to be tested.
Machine learning has emerged as a powerful approach among data-driven methods, and has been applied to study a wide range of problems in materials physics, such as material screening [5][6][7], constitutive modeling [8][9][10], scale bridging [11,12], and system identification [13,14]. Interested readers are directed to Refs [15,16] for more datadriven examples in the field of materials physics. Computational homogenization is yet another successful application of machine learning, where attempts to predict effective material properties [17][18][19][20][21] and non-linear material response [22][23][24][25][26][27][28] based on both experimentally and numerically generated data have been made by exploring different datadriven techniques. For example, convolutional neural networks (CNNs), which take images of microstructures as inputs, have been used to construct microstructure-property linkages [17] and predict macroscopic properties, such as effective ionic conductivity in ceramics [20], effective mechanical properties in composites [19] and shale [18], effective thermal conductivity in composites [21], and many others. Artificial neural networks (ANNs)/deep neural networks (DNNs), which are trained to construct complex nonlinear relationship between predefined features (e.g. strain components/volume fraction) and some quantities of interest (e.g. averaged stress responses/averaged elastic modulus ), have been coupled with finite element simulations to accelerate multiscale homogenization for bone remodeling [22], nonlinear elastic composites [27], graphene/polymer nanocomposites with nonlinear anisotropic electrical response [28], geological materials with multi-porosity [26], oligocrystals with plastic response [25], and many others. Data-driven computational homogenization has demonstrated the potential to drastically reduce computational time in traditional multilevel calculations, making possible the inclusion of detailed microstructure information in multilevel calculations [27][28][29].
In this work, a data-driven homogenization approach is explored to jointly predict the mechanical free energy and homogenized stress-strain response of a family of 2D multi-component crystalline microstructures that are numerically generated based on the computational framework in [1]. The physics underlying mechano-chemical spinodal decomposition delivers families of microstructures that are not at thermodynamic equilibrium. As outlined above, these microstructures evolve driven by the free energy. There is a hierarchical nature to the free energy of this class of material phase transformations: The strain excursions imposed on a microstructure must remain "small" in order to prevent further evolution of the microstructure itself, or the elasticity equations drive the free energy out of local basins. The corresponding structural rearrangements could then be large enough that the microstructure itself changes, leaving ambiguous the notion of homogenization. Consequently, the fluctuations in elastic free energy themselves remain small, induced by the small strains. Thus, the free energy of each microstructure has a multi-resolution structure with a dominant trajectory from microstructure-evolving phase transformations and a small-scale fluctuation value from strains exploring a given microstructure. The dominant trajectory strongly depends on the microstructural information, such as the volume fraction, the location and orientation of each crystalline phase, or the interfaces. Knowledge based neural networks (KBNNs), [30,31], which are built upon pre-trained DNNs or CNNs, are used to capture this multi-resolution data structure, with DNNs or CNNs being trained to describe the dominant part of the free energy. It is important to mention that although the term DNNs refer to a large family of neural network structures, they will specifically refer to deep neural networks with fully connected layers in this work. Our studies demonstrate that multi-resolution neural networks using both DNN-based and CNN-based KBNN models can accurately learn the macroscopic mechanical behavior of a single microstructure. Furthermore, CNN-enhanced KBNN models are capable of learning the macroscopic mechanical behavior of many microstructures from different DNS. Such KBNN models for multi-resolution learning and testing can be used to rapidly screen materials based on their microstructures for applications such as additive manufacturing, polymer blending, or material synthesis.
The rest of the paper is organized as follows. In Section 2, we summarize the mechanochemical spinodal decomposition computational framework that is used to generate different microstructures. The neural network (NN) model structures used in this work are presented in Section 3. Section 4 covers the procedures of data generation, features selection, and hyperparameter searches. The detailed simulation results are presented in Section 5. Concluding remarks and perspectives are offered in Section 6.

Mechanochemical spinodal decomposition
In this section, the computational framework to describe mechanochemical spinodal decomposition is briefly summarized. Interested readers are directed to Ref. [1] for details. e 2 =+s e e 2 =0 e 2 =-s e Figure 1: Illustration of the free energy in the strain-composition space in the low temperature phase. The chemical part of ψ has a double-well shape with respect to c, indicating a composition triggered phase transformation. The mechanical part of ψ has a convex shape at c = 0, indicating a stable square phase, and a double-well shape at c = 1, indicating a deformation triggered phase transformation of the rectangular phases.

Free energy density function
In this work, we focus on coupled diffusional/martensitic phase transformation in the two-dimensional setting. The solid has a single square phase at high temperature and undergoes a square-to-rectangle structural transformation at low temperature, which is analogous to the cubic-to-tetragonal transformation in three-dimensional space. The square lattice is the high symmetry phase that serves as the reference state for strain measurement. Here, the Green-Lagrange strain tensor E is used, 2 with its components denoted as E 11 , E 22 , and E 12 (=E 21 ). The low-symmetry rectangular lattices are derived from the square lattice by homogeneous strain. For describing the structural changes, it is more convenient to introduce three reparameterized strains, which are based on the components of E and defined as e 1 = (E 11 + E 22 )/ √ 2, e 2 = (E 11 − E 22 )/ √ 2, and e 6 = √ 2E 12 . Here, e 1 and e 6 represent the dilatation and shear strain, respectively, in the infinitesimal strain regime. The reparameterized strain e 2 uniquely distinguishes the square lattice (when e 2 = 0) and its two rectangular variants: the "positive" rectangle (e 2 > 0) with elongated lattice in the global X 1 direction and the negative rectangle (e 2 < 0) with elongated lattice in the global X 2 direction. It thus serves as a structural order parameter. The composition c, which varies between 0 and 1, is the order parameter controlling the chemistry, with c ∼ 0 denoting the composition state with the stable square phase and c ∼ 1 denoting the composition state with two unstable rectangular phases, as illustrated in Fig. 1.
At low temperature, the coupled diffusional and structural phase transformation is triggered by instabilities with respect to both the compositional parameter c and the structural order parameter e 2 . This coupled phase transformation can be described by a non-convex free energy density function ψ defined in the strain-composition space, as illustrated in Fig. 1, ψ(c, e, ∇c, ∇e) = F (c, e) + G (c, e, ∇c, ∇e), with F representing a homogeneous contribution from both composition and strain and G as a gradient-dependent non-uniform contribution to regularize the free energy density. In (1), u is the displacement field and e is a vector with e 1 , e 2 , and e 6 as its components. In the DNS, the following specific form of ψ is used to generate two-dimensional microstructures where d c , d e , s e , κ, λ e , and l e are material parameters. The free energy density function ψ in (2) consists of three contributions: a pure chemical contribution (2a), a purely elastic contribution (2b), and a mixed contribution from both chemistry and elasticity (2c). 2 Recall that E = 1 2 (F T F − 1), where the deformation gradient is F = 1 + ∂u/∂X, and u is the displacement vector.

Governing equations
Based on a generalized, Landau-type free energy density function in (1) that couples strain and composition instability, mechano-chemical spinodal decomposition can be described by a set of equations that couple the classical Cahn-Hilliard formulation and nonlinear gradient elasticity. The non-equilibrium chemistry in this coupled system is governed by ∂c ∂t where L is a transport tensor related to mobility. In (3), µ is the chemical potential, which is obtained as a variational derivative of (1) Mechanical equilibrium in the setting of strain gradient elasticity is governed by [1,2,[32][33][34][35] (most transparently written in coordinate notation): where P and B are the stress tensors, conjugate to the deformation gradient F and the gradient of the deformation gradient ∇F , respectively, whose forms are given as With appropriate initial conditions and boundary conditions, the composition and deformation fields are obtained by solving equations (3) and (5). Our implementation is with the echanoChemIGA code, which is a publicly available and highly parallelized multiphysics code developed based on PETSc [36,37], Trilinos [38,39], and PetIGA [40] libraries within the Isogeometric Analysis (IGA) framework.

Homogenized mechanical properties for heterogeneous microstructures
The microstructures obtained from solving (3) and (5) are highly heterogeneous, as illustrated in Fig. 4. To describe their macroscopic mechanical responses, the averaged deformation gradient F avg and the total mechanical free energy Ψ mech are used, which are computed as with Ω representing the domain of interest. In (7), ψ mech (c, e, ∇e) is the total elastic free energy that consists of the purely elastic term and the mixed term in (2) as The macroscopic first Piola-Kirchhoff stress tensor P avg is computed as by averaging the surface traction components (T i = P iK N K ) on a given surface Γ with normal N in the positive/negative J th direction [24].

Neural networks
In this section, the architectures of DNNs, CNNs, and KBNNs used in Section 5 are briefly discussed.

DNN
A DNN consists of multiple layers with one input layer, one output layer, and several hidden layers in between.
The inputs and outputs are called features and labels, respectively. The optimal architecture of a DNN for a specific problem is unknown a priori. Users need to select the type and structure of each layer and the number of hidden layers. In this work, DNNs specifically refer to neural networks made of fully connected (FC) layers, to distinguish from CNNs discussed in Section 3.2. A FC layer consists of multiple neurons, which take a group of weighted values and a bias as inputs, and return the output by applying an activation function to their summation. In DNNs, the weights and biases are variables subject to global optimization. The architecture of DNNs is determined by the total number of hidden layers and the number of neurons per layer, which are referred to as "hyperparameters".

CNN
CNN is a versatile type of neural network developed originally to analyze image data for tasks such as pattern detection or feature selection [41]. As discussed in the introduction, it has recently become a very useful tool for the study of study material microstructure-property relationships in situations where data from both experiments and computational material physics simulations are available as easily visualizable images. A CNN often is a mixture of convolutional layers, pooling layers, and FC layers. It can significantly reduce the dimensionality of the representation. A CNN typically requires far fewer variables than a DNN with only FC layers does for the same task. The structure of a convolutional layer is defined by hyperparameters, such as the size and number of filters, choices of paddings, and the stride numbers. In a convolutional layer, the biases and the kernel of filters are variables subject to global optimization. A pooling layer has the filter size, paddings, and stride number as hyperparameters but with no global variables.

KBNN
A knowledge-based neural network (KBNN) utilizes information from pre-trained models, as illustrated in Fig. 2. Whether or not to use a KBNN depends on the nature of the available data. For example, when the available data include abundant, less accurate data as well as expensive, scarce, highly accurate data, one can use the so-called multifidelity model. A low-fidelity model can first be trained with less accurate data, and a KBNN that is built upon the pre-trained low-fidelity model is used to improve the overall accuracy with high-fidelity data [31]. Such an approach can significantly reduce the required amount of expensive and high-fiedlity data, but still achieve the desired model accuracy. The data itself may also have a multi-resolution structure, for which one neural network may be incapable of capturing all the information. In such a scenario, one NN can be trained first to describe the dominant feature of the data. Next, a KBNN can be built upon this pre-trained model with other free variables to be trained on the same dataset. The additional variables are used to resolve other details in the data, not well-delineated by the pre-trained model. In this work, the main neural network of the KBNN is named the master neural network (MNN), and the pre-trained neural network is called the embedded neural network (ENN). The variables in the master neural network need to be optimized, whereas those in the embedded neural networks are untrainable. In another word, variables in ENN are fixed while training the MNN. Remark 1: For NNs, their global parameters are optimized via a back-propagation algorithm during the training process to drive down a loss function. The hyperparameters, which define the optimal architecture of NNs, need to be chosen by a separate process that usually involves cross-validation. For a given NN architecture, one further needs to adjust the learning rate to obtained the optimal weights and biases. A full-fledged discussion on avoidance of model underfitting or overfitting is beyond the scope of this work.

Remark 2:
The open source library TensorFlow [42] is used to create different neural network structures in this work. When NNs are used to learn a mathematical relationship with a unique physical meaning, the NNs are considered accurate only when both the label(s) and other physically meaningful quantities, usually involving the derivatives of the label(s), are accurate. For example, a DNN with fully connected layers is trained to learn the free energy density function of a Neo-Hookean hyperelastic material in [24]. For such problem, a NN is required not only to accurately represent the free energy function, but also its derivatives with respect to its features. In that specific problem, the features are the strain components and the derivatives of the NN are the stress fields. In this work, we evaluate the performance of NNs primarily based on the loss function, but also consider their derivatives whenever it is necessary. The standard automatic differentiation API from TensorFlow is utilized to compute the derivatives of NNs.

Data generation, feature selection, and hyperparameter search
In this section, we first present detailed simulation procedures to generate synthetic microstructures based on the computational framework presented in Section 2. Next, several pre-defined features for DNNs used in Section 5 are discussed. The hyperparameter search procedure for DNNs, CNNs, and KBNNs is covered in Section 4.4.

Microstructure generation
In this work, a two-dimensional solid in a domain of Ω = (0, 0.01) × (0, 0.01) with a mesh size of 60 × 60 is studied. Initially, the solid is at high temperature and has a single square phase with a randomly fluctuating composition in the range of c = 0.46 ± 0.05. A steady biaxial Dirichlet-type loading is applied to the solid, as shown in Fig. 3(a). The solid is quenched to a low temperature state with a non-convex free energy density, as given in (2), under which mechanochemical spinodal decomposition occurs.
DNS of multiple phase evolution are performed, with each of them starting from different initial compositions and mechanical boundary conditions. Throughout each DNS of phase evolution, mechanical boundary conditions remain unchanged, and the total free energy of the solid and its mechanical part are driven by the second law of thermodynamics. 3 Results from one of the many DNS are shown in Figs. 3(b) and 5(a). Selected snapshots of the composition c and the strain order parameter e 2 at different states from this particular simulation are shown in Fig. 4, in which the coexistence of the square phase, the positive rectangle phase, and the negative rectangle phase is observed. Each of the DNS takes many hundreds of time steps. We call the solutions at each time step as a frame. The homogenized deformation gradient F avg in (7), homogenized first Piola-Kirchhoff stress P avg in (9), and total mechanical free energy ψ mech in (7) are computed for each frame of every DNS. Since each frame has a different volume ratio and different spatial distribution of these three phases, it is considered as a unique microstructure, whose effective mechanical behavior differs from those of the other microstructures. Thus, each DNS will generate multiple microstructures. We discard the first 50 frames of each simulation, as distinct phase separation is not yet fully developed at this stage. In this work, 20 DNS are performed with 17000 microstructures being generated.

Data preparation
To evaluate how the macroscopic mechanical behavior of solids is related to their microstructures, 9 microstructures are uniformly sampled from each DNS with 180 microstructures being sampled in total. Combinations of different random shear and biaxial mechanical loadings are applied to each sampled microstructure. The newly applied mechanical testing loadings are much smaller than the initially applied ones for microstructure generation; hence the microstructures themselves are not altered during this posterior testing procedure. The quantities F avg , P avg , and ψ mech , are collected for each test.
Four datasets are created in this work. Datasets D I and D II , which contain microstructure features defined in Section 4.3, the e 2 solution and the ψ 0 mech from DNS, are created for microstructures from a single DNS and all microstructures from different DNS, respectively. Datasets D III and D IV contain mechanical testing information for a single microstructure and all the sampled microstructures, respectively. Specifically, in dataset D III , the microstructure at frame 400 from one particular DNS, as shown in Fig. 5(a), is tested with 1600 different combinations of mechanical loading. The elastic free energy ψ mech from all the 1600 tests is plotted in Fig. 5(b), where ψ mech is oscillating around a base elastic free energy ψ 0 mech = −0.01923. Here, ψ 0 mech refers to the elastic free energy stored in the microstructure during phase evolution shown in Figs 3b and 4, which is before the mechanical tests. The magnitudes of oscillations in ψ mech in Fig. 5(b) further confirms that the magnitude of the applied mechanical loadings are very small. In dataset D IV , all the 180 sampled microstructures, 9 of which from one specific DNS are shown in Fig. 5(a), are tested under different mechanical loadings with 57600 data points collected.

Microstructure feature selection
To differentiate microstructures from each other, several features are selected. These features include volume fractions φ + r and φ − r for the positive and negative rectangle phases. The volume fraction of the square phase is not selected as an independent feature because it can be calculated as φ s = 1 − φ + r − ψ − r . Other selected features include the interfacial length between the square phase and the rectangle phases l r s , as shown in Fig. 6(a), the interfacial length of the positive rectangle phase l r+ , as shown in Fig. 6(b), and the interfacial length of the negative rectangle phase l r− , as shown in Fig. 6(c).

Hyperparameter search
As discussed in Section 3, the optimal architecture of NNs is unknown a priori. Hyperparameters can be selected via either manual tuning or automatic optimization algorithms, such as grid search or random search [43]. In this work, grid search is performed for all the NNs. For DNNs and the MNN of KBNNs, we search for the number of hidden layers (N HL ) and the number of neurons per layer (N NPL ). In our search space, N HL varies between 1 and 10 with a step of 1. An identical N NPL is assumed for each hidden layer with its value varying between 2 and 256 for a step of 2. For CNNs, a kernel size of (3, 3) and a stride size of (2, 2) are pre-chosen. We only search for N HL and the number of filters per layer (N FPL ), with N HL varying from 1 to 10 for a step of 1 and N FPL varying from 2 to 32 for a step of 1. Unlike the case of N NPL for DNNs/MNNs, N FPL is not identical for each layer. Its value increases with the depth of the hidden layer. In this process, the exponentially decaying learning rate implemented in Tensorflow, which follows a stair case function, is used with an initial learning rate lr 0 = 0.001, a decay rate v decay = 0.7, a decay step N decay = 100, and a final N total = 2000 epochs. The dataset is randomly split into a set consisting of 90% for training and validation and a set of 10% for testing. K-fold cross-validation procedure (with k = 5) [43] is performed on the set consisting of 90% of the data to train and evaluate different NN models. Feature normalization and label scaling are used to improve the accuracy of NNs during training.
When performing the hyperparameter search, first, the total number of variables of each possible NN architecture in our search space is computed and sorted in an ascending order. These NNs with a total variable number larger than the size of the dataset will be excluded from the search space. Then, a grid search based on the total number of variables of the NNs is performed. The performance of each NN is evaluated based on the averaged validation loss and is sorted in an ascending order. The total number of variables of the top 30% performing NNs defines a refined search space, in which a new grid search is performed. The grid search is repeated three times in total. The model with the smallest averaged validation loss is selected as the best one. The hyperparameter search procedure is summarized in the Algorithm Box 1.
1: Create a set S containing all possible NN structures that lies in the search space defined by hyperparameters (N HL , N NPL , or N FPL ), with NNs in S being sorted in an ascending order based on the total number of variables (V total ) of each NN. 2: Grid search of hyperparameters in S based on V total . 3: Define an initial lower limit and an initial upper limit of V total with V min total = 0 and V max total = size of (dataset D). 4: for s in multiple sampling steps (= 3, in this work) do 5: Uniformaly sampling multiple (= 25, in this work) NNs out of all NNs, where each NN has V min total ≤ V total ≤ V max total , to form a subsetS. 6: Perform K-fold cross-validation for each NN inS. 7: for each model M i inS do 8: Split D into K mutually exclusive subsets D k 9: for k from 1 to K(= 5, in this work) do Sort models inS based onL i in an ascending order. 16: Refine the search space by updating V min total and V max total , where V min total = min(V total ) and V max total = max(V total ) in S 30 , withS 30 representing a subset ofS that contains the top 30% (an user-defined threshold value) performed models. 17: end for 18: Select the best model M with the smallestL.

Numerical examples
In this section, we explore different NNs to predict the homogenized mechanical behavior of synthetically generated heterogeneous microstructures. Specifically, the base elastic free energy of microstructures from multiple DNS is studied in Section 5.1 with both CNNs and DNNs. The homogenized mechanical behavior of a single microstructure is studied with KBNNs in Section 5.3. Finally, CNN-enhanced KBNNs are trained to predict the homogenized mechanical behavior of different microstructures from multiple DNS in Section 5.4.

Base mechanical free energy for one DNS
As revealed in Figs. 3 and 5, the elastic free energy ψ mech stored in microstructures due to phase evolution is of a sharply multi-resolution nature. It has ψ 0 mech from microstructure phase evolution as the dominant feature and ∆ψ mech from mechanical testing as the detailed feature. It is challenging to capture both features by a single NN, because the weights emphasize the dominant feature over the detailed feature during the training process. To overcome this challenge, we use KBNNs, as discussed in Section 3.3, to represent this multi-resolution data. The ENN is trained to learn the base free energy ψ 0 mech (D I ) in this section with both DNNs and CNNs being explored.

CNN
The microstructure features selected in Section 4.3 are the interpretation of image data based on authors domain knowledge of the global quantities that distinguish microstructures. Alternately, we can train CNNs to automatically identify features to represent microstructures. Such an approach underlies the treatment of this section with the goal of investigating the existence of any advantage for CNNs over DNNs for computational materials physics simulations.
A CNN consisting of multiple convolutional layers, multiple pooling layers, and one dense layer trained to predict the base elastic free energy in Fig. 5(a). The CNN takes the whole e 2 field solution from DNS as inputs, with a pixel resolution of 61 × 61, and ψ 0 mech as its label. A hyperparameter search is conducted by following the procedure discussed in Section 4.4, with the best architecture of the CNN given in Table 1 with a total variable number of 590. The model is trained with the Adam optimizer for 10000 epochs with the exponentially decaying learning rate given in (10) where v decay = 0.92. The learning curve for the CNN is plotted in Fig. 7(d). The model can accurately predict ψ 0 mech , as plotted in Figs. 7(e,f), which show an improved accuracy compared with the DNN results in Figs. 7(b,c).

Base elastic free energy for multiple DNS
In this section, both DNNs and CNNs are explored to represent the base free energy ψ 0 mech (D II ) from multiple DNS for the ENN. As in Section 5.1.1, the DNN takes φ + r , φ − r , l r s , l r+ , and l r− as its features and ψ 0 mech as its label. The results of an optimal DNN structure obtained from the hyperparameter search, which has N HL = 7, N NPL = 48, and V total = 14449, are shown in Fig. 8(a-c). The results of an optimal CNN structure, whose architecture is given in Table 2 with V total = 4403, are shown in Fig. 8(d-f). From Fig. 8, one can observe that both the DNN and the CNN show a good representation of the base free energy from multiple DNS with different initial conditions and boundary conditions.

Homogenized mechanical behavior of single microstructure
In this section, KBNNs are constructed to study the homogenized mechanical behavior of a single microstructure (dataset D III ), with ENNs being either pre-trained DNNs or CNNs. The ENNs offset the dominant feature from the datasets to allow KBNNs to capture the detailed feature. This is achieved via a new MSE loss function with the form where Y is the label, Z is the KBNN predicted value, ψ mech is the DNS value of the elastic free energy after mechanical testing, and ψ 0 mech,NN is the ENN predicted base elastic free energy of the microstructure before mechanical testing. In (11), Y essentially represents the change of mechanical free energy ∆ψ mech due to the posterior mechanical testing.

DNN-based KBNN
With the DNN in Section 5.1.1 in hand, we now build a KBNN model with the structure presented in Fig. 2, with F 11 , F 12 , F 21 , F 22 , φ + r , φ − r , l r s , l r+ , and l r− as features and ψ mech as the label. In this KBNN, the embedded pre-trained DNN takes {φ + r , φ − r , l r s , l r+ , l r− } to predict ψ 0 mech,NN . The remaining features {F 11 , F 12 , F 21 , F 22 } and the shifted label ∆ψ mech = ψ mech − ψ 0 mech,NN are used to optimize the variables of the MNN. The MNN is not exposed to the features {φ + r , φ − r , l r s , l r+ , l r− }, and therefore does not have information on the microstructure that it is training against. This is a refinement we undertake in Section 5.4. The optimal values of N HL and N NPL for the MNN are searched by following the procedures in Section 4.4. An L 2 kernel regularization with a factor of 0.001 is applied to the input layer to minimize the coefficients of less important features to reduce overfitting. The Softplus activation function is used. An optimal MNN is obtained with N HL = 3, N NPL = 24, and V total = 1345. The KBNN is trained with the Adam optimizer for 10000 epochs with the exponentially decaying learning rate given in (10) where v decay = 0.92. The learning curve for the KBNN is plotted in Fig. 9(a), where neither overfitting nor underfitting is observed. Fig. 9(b) shows that the KBNN can capture the detailed features of the data and predict ∆ψ mech with satisfactory accuracy. The derivative of ∆ψ mech, NN with respect to F are shown in Fig. 9(c-f), where the KBNN (c-f) the components of P KBNN vs P DNS , where the KBNN shows good performance on P 11 and P 22 , but not P 12 and P 21 due to the fact that P 12 and P 21 are one order of magnitude smaller than P 11 and P 22 .
shows good performance on P 11 and P 22 , but not P 12 and P 21 due to the fact that P 12 and P 21 are one order of magnitude smaller than P 11 and P 22 in the DNS.

CNN-based KBNN
A CNN-based KBNN is built with F 11 , F 12 , F 21 , F 22 , and the image of the e 2 field solution as features and ψ mech as the label. In this KBNN, the embedded pre-trained CNN takes the image of the e 2 field solution of the base microstructure to predict ψ 0 mech . The remaining features {F 11 , F 12 , F 21 , F 22 } and the shifted label ∆ψ mech = ψ mech − ψ 0 mech are used to optimize the variables of the MNN. The identical MNN as in Section 5.3.1 is used and trained. Fig. 10(b) shows that the KBNN can capture the detailed features of the data and predict ∆ψ mech with satisfactory accuracy. The derivative of ∆ψ mech, NN with respect to F are shown in Fig. 10(c-f), similar as the results in Fig. 9, where the CNN-based KBNN also shows good performance on P 11 and P 22 , but not P 12 and P 21 .

Homogenized mechanical behavior of microstructures from multiple DNS
Expanding beyond the studies for a single microstructure, KBNNs are constructed to predict the homogenized behavior of multiple microstructures from different DNS (dataset D IV ). KBNNs similar to those used in Section 5.3 are investigated. However, the MNN with {F 11 , F 12 , F 21 , F 22 } as features is incapable of describing the homogenized mechanical behavior of different microstructures, as such a simple MNN is unaware of the details of each microstructures. Our studies also confirm that even the inclusion of pre-defined microstructure related features {φ + r , φ − r , l r s , l r+ , l r− } to the MNN shows insignificant improvement of the performance of KBNNs for multiple microstructures.
Since the MNN with pre-defined features has insufficient expressivity to describe the homogenized mechanical response across microstructures, a CNN-enhanced KBNN structure, as shown in Fig. 11, is explored. Here, the CNN (c-f) the components of P KBNN vs P DNS , where the KBNN shows good performance on P 11 and P 22 , but not P 12 and P 21 due to the fact that P 12 and P 21 are one order of magnitude smaller than P 11 and P 22 . enhancement is utilized to identify the most relevant features from the e 2 fields. A manual hyperparameter tuning is performed. The details of an MNN with satisfactory performance, which has a total variable number of 9297, are summarized in Table 3. Our results, as shown in Fig. 12, confirm the effectiveness and good performance of the new KBNN structure, which can accurately predict the mechanical free energy on the test dataset. Furthermore, the P 11 and P 22 components of P KBNN , obtained by taking derivatives of the KBNN with respect to the deforma-  Figure 11: Illustration of the structure of CNN-enhanced KBNN. The ENN, which either takes pre-defined features or microstructure images, is used for offset the dominant feature. The CNN in the middle, which takes the perturbed e 2 field information, is used to identify the most relevant features for homogenized mechanical behavior prediction. The combination of the outputs from the CNN and the deformation gradient F components serve as the input for a fully connected DNN for resolving the detailed features of the dataset. (c-f) the components of P KBNN vs P DNS , where the KBNN shows good performance on P 11 and P 22 , but not P 12 and P 21 due to the fact that P 12 and P 21 are one order of magnitude smaller than P 11 and P 22 .
tion gradient F , match well with respective components of P DNS . The new KBNN structure which performs well at learning the homogenized mechanical behavior of different microstructures demonstrates the advantage of utilizing CNNs in a multi-resolution learning framework for this instance of computational material physics applications, with heterogeneous microstructures.

Conclusions
In this work, different NN architectures are used to study the homogenized mechanical behavior of microstructures generated by mechano-chemical spinodal decomposition. Our preliminary results show the promise of applying CNNs in computational material physics. Particularly, we have demonstrated that both a CNN-based KBNN and a CNNenhanced KBNN can be trained to rapidly predict the elastic response based on the images of microstructures.
Our investigations toward infusing the better-performing CNN architectures with interpretability reveal that the convolutional layers isolate a greater number of microstructural features than those that we identified on the basis of domain knowledge: {φ + r , φ − r , l r s , l r+ , l r− }. The volume fraction and interfaces appear as recognizable outputs from more than two and three convolutional layers, respectively. While not presenting a set of features with the parsimony that the expert may postulate for the problem, it suggests that CNN architectures use redundancy to outperform DNNs. Interestingly, it also raises questions about the completeness of the feature set {φ + r , φ − r , l r s , l r+ , l r− } that was imposed on the DNN model, suggesting that there are epistemic gaps in the experts' understanding of this problem. This is important for future studies on combining image data from experiments with multiphysics simulations. Although this work focused on two-dimensional simulations, our results point to the CNN being more effective in the three-dimensional study. Because 3D data is more complex in its information content, our domain knowledge might harbor further inadequacies to identify the relevant features. The CNN, instead, could prove more effective at feature selection and dimensionality reduction.