We recently proposed the adaptive incrementally affine method a s a novel homogenization scheme that enables accurate prediction of the nonlinear mechan ical response of particulate- reinforced composites under cyclic or multiaxial loading condit ions. However, being a mean- field homogenization method that relies on the solution of Eshe lby’s inclusion problem, the aforementioned method does not provide accurate predictions whe n the shape of the reinforcement has a high aspect ratio or the interaction among reinforcements becomes significant. Herein, we propose a combined theoretical and data -driven approach in which the homogenization method is followed by transfer learning to enhan ce the prediction of the nonlinear mechanical response of particle/fiber-reinforced comp osites. We trained a deep neural network (DNN) with a massi ve stress-strain curve dataset of a composite subjected to uniaxial loading in the elasto-plastic regime based on the adap tive incrementally affine homogenization method; then, we fine-tuned the pre-trained DNN via transfer learning with a relatively small dataset based on time-consuming three-dimensio nal (3D) finite element analyses (FEA). The transfer lea rning approach exhibited better predictive performance than the DNN directly trained only with the FEA dataset did, for a w ide range of reinforcement volume fractions and shapes. With transfer learning, the DNN ex ploits the knowledge learned from the homogenization theory to perform a new learning task o n a small dataset from 3D FEA. The combined theoretical and data-driven approach proposed herein can be extended to other composite analyses when experiments or simulations requir e excessive time and cost for a large dataset and the applicab ility of the homogenization the ory is limited. 3 1. Introduction With the addition of various forms of reinforcements, reinforce d composites have evolved to outperform single-phase materials. Predictions of th e effective responses of such composites are crucial for efficient machine element design and structural optimization across a variety of industries. As the first generation of composite a nalysis, homogenization methods based on analytical formulations have been investigated for man y decades [1, 2]. These methods entail lower computational costs compared with finite e lement analysis (FEA) or experiments but have limitations owing to the use of mathematic al assumptions. For example, the mean-field homogenization theory is valid under a relativel y small volume fraction of reinforcement (<20%) because the mean-field assumption is viola ted as the volume fraction increases [3-6]. If a homogeni zation method relies on the solut ion of Eshelby’s inclusion problem, reinforcements with a non-ellipsoidal sh ape can only b e approximately treated [7-9]. As the second generation of composite analysis, the finite elem ent method (FEM) has been extensively employed with the advancement of computational capa bility [10]. However, the nonlinear behavior of materials and structural complexity signi ficantly increase the computational cost of the FEM, which limits its applicability. Recently, as the third generation of composite analysis, data-based machine learning [11, 12], wh ich incorporates machine learning techniques into the com posite analysis [13-16], has dr awn significant attention. Once a machine learning model has been trained, it can immediately p redict the outputs for given input values; however, it require s a large amount of data for r eliable prediction. Each method has advantages and disadvantages, and thus, a singl e method would not be a satisfactory or universal route toward efficient composite analyses. For example, the homogenization theory enables exc eedingly fast prediction of th e effective properties of composites, but its accuracy and applicability are limited. Mea nwhile, both three-dimensional 4 FEM analysis and direct experimental measurements offer highly accurate predictions of composite properties but at the expense of excessive time or co st. However, the accuracy and efficiency of composite analysis can be significantly improved by training a machine learning model with a large but less accurate dataset based on the homog enization method and then improving the machine learning model with an accurate but small dataset using the FEM or through experiments. Herein, we propose a combined theoretical and data-driven framework to realize such a hybrid composite a nalysis via transfer learni ng. Various mean-field homogenization methods have been proposed ba sed on Eshelby’s solution [1]. For linear elastic composites, the Mori–Tanaka [2 , 17], differential [18, 19], and double-inclusion [20] methods have been widely used. For compos ites exhibiting nonlinearity, such as viscosity or plasticity, the incrementally affine [21, 22], incrementally secant [23, 24], and adaptive incrementally affine [25, 26] methods have been pr oposed. These methods are reliable when the shape of the reinforcement is close to a sphe re and the volume fraction of the reinforcement is small. However, many technologically important composites are synthesized with a variety of forms and a relatively large volume fraction of reinforcement, which limits the applicability of homogenization methods. In contrast, deep learning techniques have been employed to predict the mechanical properties of two-dimensiona l checkerboard composites [15, 27-29] or three-dimensional linear elastic composites [30, 31]. It is well known that with more data, the generalization of a deep neural network (DNN) mo del is improved for unseen data. When a sufficiently larg e training dataset does not exist , the performance of DNNs becomes inferior to that of conventional machine learning techn iques such as linear regression, support vector machine, or decisi on tree-based algorithms [16]. However, it is often difficult to obtain sufficiently large data sets owing to time and cost pr oblems. Given that homogenization methods have limitations in terms of accuracy and that 5 DNNs require excessively large amounts of data, we propose inco rporating transfer learning techniques to composite analysis. Transfer learning is a method that uses a model trained once for a particular task as a star ting point for training a second model [32]. Therefore, transfer learning is used when it is difficult to obtain sufficient data [33-37]. The representative volume element (RVE) of a composite with reinforcements having a high aspect ratio in the nonlinear regime entails substantial comput ational costs associated with FEM; then, it becomes difficult to obtain the amount of data re quired for machine learning. In the present study, we show that a framework combining the homogenization theory and transfer le arning can predict the effective behavior of the elasto-plastic composite efficiently and reliably; a sufficiently large dataset is generated using the homogenization theory for the in itial DNN training, at a low computational cost; subsequently , the pre-trained DNN is fine-t uned using relatively small FEM data obtained via transfer learning. The workflow of our tr ansfer-learning framework is illustrated in Fig. 1. The remainder of this paper is organized as follows. The consti tutive equations for modeling the elasto-plastic behavior are summarized in Section 2.1. A description of the data generation is presented in Section 2.2. The adaptive incrementa lly affine and yield reduction methods are briefly introduced for the homogenization data. The face-centered-cubic (FCC) structured RVE is introduced for FEM data. The procedure for ap plying transfer learning is d e s c r i b e d i n S e c t i o n 3 . T h e r e s u l t s f o r t h e e l l i p s o i d a l p a r t i c l e-reinforced composite are presented in Section 4.1. The results for the fiber-reinforced composites are included in Section 4.2. The conclusions are presented in Section 5. Boldface letters indicate second- or fourth-order tensors and n on-bold face letters indicate scalar values; (:) denote s a double contraction, as 𝒂: 𝒃 ൌ 𝑎 ௜௝𝑏௝௜; ሺ𝑨: 𝒃 ሻ௜௝ൌ𝐴 ௜௝௞௟𝑏௟௞. 6 2. Material modeling and numerical method 2.1. Material modeling We consider composites containing an elasto-plastic matrix. We note that the methodology employed in this study can be applied to a variety of other composite materials as well that show viscoelastic or viscoelastic-viscoplastic res ponses. In the elasto-plastic constitutive model, the total strain ( 𝜺) is divided into t he elastic part ( 𝜺𝒆) and plastic part ( 𝜺𝒑): 𝜺ൌ𝜺𝒆൅𝜺𝒑. (1) Then, the Cauchy stress ( 𝝈ሺ𝑡ሻ) is determined as follows: 𝝈ሺ𝑡ሻൌ𝑪 : 𝜺𝒆ൌ𝑪 : ሺ 𝜺െ𝜺𝒑ሻ. (2) Assuming isotropic materials, the fourth-order stiffness tensor (𝑪) is expressed as, 𝐶௜௝௞௟ൌ𝐸𝜈 ሺ1 െ 2𝜈ሻሺ1 ൅ 𝜈ሻ𝛿௜௝𝛿௞௟൅𝐸 2ሺ1൅𝜈 ሻ൫𝛿௜௞𝛿௝௟൅𝛿 ௜௟𝛿௝௞൯, (3) where 𝐸 is Young’s modulus, 𝜈 is Poisson’s ratio, and 𝛿 is the Kronecker delta. Rate-independent isotropic harde ning is assumed for plasticity. The yield function ( 𝑓) is defined as follows: 𝑓൫𝜎௘௤,𝑝 ൯ ≡𝜎 ௘௤െቀ 𝜎 ௬൅𝑅ሺ𝑝ሻቁw h e r e 𝜎 ௘௤ൌඨ3 2𝒔: 𝒔 (4) 𝑝ൌ ඨ2 3𝜺𝒑:𝜺𝒑, (5) where 𝒔 is the deviatoric part of the Cauchy stress, 𝜎௘௤ is the von Mises stress, 𝜎௬ i s t h e yield stress, 𝑅 is the hardening stress, and 𝑝 is the accumulated plastic strain. For hardening responses beyond the yield, isotr opic hardening with the power law is adopted: 𝑅ሺ𝑝ሻൌ𝑘 𝑝௡ (6) where 𝑘 is the hardening modulus, and 𝑛 is the hardening exponent. Thus, isotropic elasto- 7 plastic behavior can b e modeled with five parameters: Young’s m odulus, Poisson’s ratio, yield stress, hardening modulus, and hardening exponent. 2.2. Data generation with a homogenization method and 3D FEM mo deling C o n s i d e r i n g t h a t t h e r e i n f o r c e m e n t i s m u c h s t i f f e r t h a n t h e matrix materials in most composites, the composite is assumed to be an elasto-plastic ma trix reinforced with linear elastic particle/fiber. Its composition can be characterized by nine parameters related to the elastic properties (Young’s modulus 𝐸଴, Poisson’s ratio 𝜈଴) and plastic properties (yield stress 𝜎௬, hardening modulus 𝑘, hardening exponent 𝑛) of the matrix, as well as the elastic properties (Young’s modulus 𝐸ଵ, Poisson’s ratio 𝜈ଵ), aspect ratio (AR), and volume fraction (𝑓ଵ) of the reinforcements. After normalizing 𝐸ଵ,𝜎௬, and 𝑘 with 𝐸଴, the input parameters can be reduced to a set of eight dimensionless parameters. With the normalized eight parameters as the input, we set the effective stress-strain cur ve of the composite subjected to uniaxial loading as the output. We consider a wide range of inp ut parameters to cover a variety of elasto-plastic composite behavi ors, as presented in Table 1. T h e a d a p t i v e i n c r e m e n t a l l y a f f i n e m e t hod with yield reduction is adopted as the homogenization method because it has been shown to offer highly accurate predictions of the complex nonlinear mechanical response of particulate-reinforced composites beyond the elastic regime, under cyclic and multiaxial loading. In the reg ime where the assumption of micromechanics-based mean-field homogenization theory holds, th a t i s , a s m a l l v o l u m e fraction of a reinforcement having an ellipsoidal shape with an aspect ratio close to 1, the adaptive incrementally affine method offers highly accurate pre dictions; however, its accuracy is diminished with an increase i n the reinforcement volume frac tion or aspect ratio. 8 The complete details of the homogenization method can be found in our previous study [25, 26]; we provide some of these details herein for clarity. The yield reduction method reduces the yield stress of the matrix with the volume fraction of the reinforcement. 𝜎୷←ሺ1െ𝑣 ଵሻ𝜎୷for the matrix (7) The adaptive incrementally affine method for elasto-plastic com posites is summarized as follows. Here, subscripts 0 and 1 represent variables of the ma trix and reinforcement, respectively (e.g., 𝑪𝟎 refers to the stiffness tensor of the matrix and 𝑪𝟏 refers to the stiffness tensor of the reinforcement). 1. A macro-strain increment ( ∆𝜺̅) is applied to the composite for a given strain state at 𝑡௡. 2. The strain increment of the r einforcement is initialized as a macro-strain increment: 〈∆𝜺〉୵భൌ∆ 𝜺 ̅ where 〈•〉୵బ and 〈•〉୵భ refer to the volume averages over the matrix and particles, respectively. 3. The strain increment of the m atrix is calculated as follows: 〈∆𝛆〉୵బൌ1 𝑣଴൫∆𝜺തെ𝑣 ଵ〈∆𝛆〉୵భ൯. (8) 4. Eshelby’s tensor ( 𝑺) [1, 38] and the concentration tensors ( 𝑩ଵ𝜺, 𝑨ଵ𝜺) are computed as follows: 𝑩ଵ𝜺ൌሾ𝑰൅𝑺 : ሺ𝑪𝟎ሻିଵ∶ሺ𝑪𝟏െ𝑪 𝟎ሻሿିଵ (9) 𝑨ଵ𝜺ൌ𝑩ଵ𝜺:ሺ𝑣଴𝑰൅𝑣 ଵ𝑩ଵ𝜺ሻିଵ (10) where 𝑰 is the fourth-order identity tensor. 5. The residual tensor ( 𝑹) is computed as follows: 𝑹ൌ 𝑨ଵ𝜺:∆ 𝜺തെ〈∆𝜺〉௪భ ൅ሺ𝑨ଵ𝜺െ𝑰ሻ:ሺ𝑪𝟏െ𝑪 𝟎ሻିଵ:ሺ𝝈𝟏ሺ𝑡௡ାଵሻെ𝑪 𝟏:𝜺𝟏ሺ𝑡௡ାଵሻെ𝝈 𝟎ሺ𝑡௡ାଵሻ ൅𝑪 𝟎:𝜺𝟎ሺ𝑡௡ାଵሻሻ ൅ሺ𝑰െ𝑨ଵ𝜺ሻ:ሺ𝑰െ𝑩ଵ𝜺ሻିଵ:ൣ 𝑩ଵ𝜺:𝑪𝟎ିଵ:𝝈𝟎ሺ𝑡௡ሻെ𝑪 𝟏ିଵ:𝝈𝟏ሺ𝑡௡ሻ൧ (11) 6-1. If |𝑹|൐𝑡 𝑜 𝑙 𝑒 𝑟 𝑎 𝑛 𝑐 𝑒 , step 3 is repeate d with a new trial. 9 〈∆𝜺〉௪భ←〈∆𝜺〉௪భ൅𝑹 (12) 6-2. If |𝑹|൏𝑡 𝑜 𝑙 𝑒 𝑟 𝑎 𝑛 𝑐 𝑒 , then step 7 is followed. 7. The macro stress increment ( ∆𝝈ഥ) and effective stiffness tensor ( 𝑪ഥ) are obtained as follows: ∆𝝈ഥൌ𝑣 ଴𝑪𝟎:൫〈∆𝜺〉௪బെ∆ 𝜺𝟎𝒑൯൅𝑣 ଵ𝑪𝟏:൫〈∆𝜺〉௪భെ∆ 𝜺𝟏𝒑൯. (13) 𝑪ഥൌሾ𝑣଴𝑪𝟎൅𝑣 ଵ𝑪𝟏:𝑩ଵ𝜺ሿ:ሾ𝑣଴𝑰൅𝑣 ଵ𝑩ଵ𝜺ሿିଵ. (14) 8. Step 1 is repeated for the next strain increment. Under a given reinforcement volume fraction, the prediction acc uracy of the homogenization method decreases as the aspect ratio of an ellipsoidal particle increases; this is because the mutual interaction among reinforcements increases, which furthe r violates the mean-field assumption. Additionally, the prediction provided by the homoge nization method has another source of error for cylindrical fiber-reinforced composites bec ause Eshelby’s solution assumes an ellipsoidal shape for the reinforcement. T h e F E M o f f e r s a n accurate numerical evaluation of the internal stress distributi on and yield inside a composite under mechanical loading, and the outc ome from the FEM can be considered as a ground truth if the implementation and numerica l method are carefully executed. We employed the commer cial software ABAQUS [39] with the UMAT code for the FEM simulations in this study. FEM data were generated consider ing the RVE; the reinforcement was embedded into an FCC configuration, as shown in Fig. 2. The analysis was conducted using only one-eighth of the RVE under periodic bound ary conditions (PBCs). Depending on the aspect ratio and volume fraction of the reinfo rcement, approximately 15,000 to 108,000 elements are required to ensure convergence. T h e c o m p u t a t i o n a l c o s t s f o r t h e h o m o g e n i z a t i o n m e t hod and FEM for the analysis of the composite are compared in Fig. 3. Here, we consider composites containing reinforcements 10 with an aspect ratio of 10, while the other input parameters ar e randomly assigned. For analyzing 100 composites, the homogenization method requires 5. 2 h of single-core computation, whereas the FEM n eeds 56.8 h of eight-core computa tion; thus, the FEM is 87.4 times more expensive than homogenization in terms of computatio nal cost. Considering a variety of input parameter sets, the predictions provided by th e adaptive incrementally affine method are compared with the FEM results for the ellipsoidal-pa rticle/fiber-reinforced composite in Fig. S1 (see Supplementary Information). Predictio ns from the homogenization methods show notable errors when the aspect ratio of the reinfo rcement or volume fraction is large. Moreover, although the FE M r e s u l t s a r e c o n s i d e r e d a s g r o und truth in this study, experimental data (which also account for the effects from vari ous manufacturing defects) can be used in the transfer learning scheme as well; this is discus sed in the following section. 3. Application of transfer learning Transfer learning has been widely used to train a task-specific model effectively by utilizing a pre-trained model with a massive dataset. Transfer learning c an achieve powerful predictive performance even with a small amount of task-specific data by f ine-tuning a subset of layers of the pre-trained model while typically freezing the feature e xtraction layers. In this study, we first pre-trained a DNN model w ith sufficient homogenization da ta to enable this pre-trained model to capture the underlying mechanism determining the effec tive stress-strain relationship for idealized ellipsoidal partic le-reinforced composites. We th en used task-specific data (FEM data) to fine-tune the pre-trained model for predicting the str ess-strain relationship for an ellipsoidal-particle/fiber-rei nforced composite. For the fine-t uning procedure, unlike in typical transfer learning, we trained the task-specific model with FEM data using the parameters of the pre-trained model as the ini tial parameters, instead of fre ezing subset layers from the pre- 11 trained model. We found that freezing some subsets degrades the performance of the fine-tuned DNN. This is because the knowledge that we seek to transfer is not an abstract feature but rather the general input and out put structural relationships th at are believed to be embedded in the overall layer of the pre-trained model. As shown in Fig. S2 , the pre-trained model represents the general structure of the str ess-strain curves well (the out puts are concave and monotonic functions). The overall workflow of the trans fer-learning fram ework is illu strated in Fig. 1. The DNN was constructed and trained using Keras [40] on Python 3.7. We first generated 59,500 data using the homogenization method. We then constructed 2,000 data from computationally expensive FEM simulations. The homogenization data were divided into training and validation sets in a ratio of 49,000:10,500. The FEM data were divided into training, validation, and test sets in a ratio of 1,400:300:300. The validation set w as used as an early stopping criterion for training the DNN, a nd the test set was used for a ssessing the model. Each input parameter listed in Table 1 is normalized to values between 0 a nd 1. The output data, i.e., the stress-strain curves evaluate d at 101 strain poi nts (i.e., the dimension of the output is 101), are compressed into 10-dimensional vectors comprising the 10 most i mportant principal components extracted via principal component analysis (PCA), as depicted in Fig. 4 (a), according to a previously proposed scheme [15]. The cumulative explained variance, which is defined as the sum of eigenvalues normalized by the sum of all eigenvalues ( ∑ 𝜆௜௜/ଵ଴ ௜ୀଵ ∑ 𝜆௜௜ଵ଴ଵ ௜ୀଵ ) is 99.999%. The data that are dimensionally reduced via PCA, i. e., the 10 most important principal components, are normalized to values between 0 and 1 for efficient training. We chose a DNN architecture, which does not impose any structural assump tion (inductive biases), to employ as flexible a model as possible to express the complex s tress-strain relationship of e l a s t o - p l a s t i c c o m p o s i t e s . A d e n se layer, batch normalization, and activation function 12 constitute a unit layer in the DNN architecture as shown in Fig . 4 (b). Batch normalization is known to accelerate training, mitigate the sensitivity to initi a l w e i g h t s , a n d h a v e a regularization effect [41, 42]. Th e weighted mean squared error (WMSE) is defined for DNN training as follows: WMSE ൌ1 𝑛෍෍ ට𝜆௝௝൫𝑌௝,௜௧௥௨௘െ𝑌௝,௜௣௥௘ௗ൯ଶଵ଴ ௝ୀଵ௡ ௜ୀଵ (15) where 𝑌௝,௜ refers to the j-th principal component of the i-th sample, 𝜆௝௝ refers to the j-th eigenvalue, and 𝑛 refers to the number of samples. The variation between eigenva lues is exceedingly high; hence, the squared eigenvalues are used as we ights. To optimize the DNN architecture, grid-search hyper-parameter tuning is performed t o search for the best combination of the num ber of unit layers, the number of neurons in each layer, and the type of activation function [43-45]. We chose the architecture with the lowest average error after five epochs in the following search s pace (see Supplementary Note 1) . Number of neurons of each layer = [40, 50, 60, 70, 80, 90, 100, 110, 120] Number of hidden unit layers = [3, 4, 5, 6, 7] Activation function = [ReLU , GeLU, ELU, LeakyReLU] The best architecture has 50 neurons for each layer, six hidden unit layers, and exponential linear unit (ELU) as an activati on function as shown in Fig. 4 (b); this leads to 14,010 trainable parameters. The Adam optimizer, an adaptive learning rate optim izer, was used for training [46]. The batch size was set to one-tenth of the training datas et size. After each epoch, the training dataset was randomly shuffled to minimize the sample b ias. As depicted in Fig. 1, Model A was trained using sufficient homogenization data with a low computational cost. Taking the weights of Model A as the initial weights, Model B w as fine-tuned using only a few FEM data, which entailed a relat ively high computational cost. Then, based on the trained 13 model, 101-dimensional stress-st rain curves were recovered from 10-dimensional principal component arrays via reverse normalization and reverse PCA. Her einafter, Model A refers to a DNN model trained with homogenization data, Model B represents a DNN model trained via transfer learning based on both homogenization data and FEM dat a, and Model C indicates a DNN model trained only with FEM data. Neural networks must have sufficient width, depth, and trainabl e parameters when the relationship between the input and output is complex. In such c ircumstances, a large amount of data is required for the DNN to achieve satisfactory general ization performance (i.e., prediction of unseen data). Training with a small amount of dat a would lead to an over-fitted model with poor generalization. The transfer learning strategy proposed in this study is illustrated in Fig. 5. The x-axis represents how general k nowledge is extracted from the abundant homogenization data t hrough pre-training. Owing to the abundance of the homogenization data, the pre-trai ned model becomes sufficiently general to serve as a structural foundation for vari ous task-specific models. The y-axis represents how the pre- trained model is fine-tuned using FEM data to form task-specifi c models. Model C suffers from overfitting owing to the lack of data, especially when using a flexible DNN architecture. Meanwhile, Model A cannot accurately predict the task-specific stress-strain relationship of composites, because of the limitation of the homogenization the ory. Despite the difference between the predictions provided by homogenization and the FEM, these predictions share the underlying mechanism determining the effective stress-strain re lationship. We believe that Model A captures the underlying mechanism based on the homogeni zation theory. Relying on the general stress-strain relati onships extracted from the pre- trained model, Model B can be quickly adapted for ellipsoidal-particle/fiber-reinforced compo sites, avoiding overfitting, even when using a small amount of task-specific FEM data. In summary , we used a large volume of 14 low-fidelity data to train a general and sufficiently large DNN model to extract the structural relationships between the stre ss and strain of idealized compos ites. We then fine-tuned the pre- trained model to increase the predictive accuracy using few but high-fidelity FEM data. 4. Result and discussion We used the DNN trained via transf er learning to predict the st ress-strain curves of elasto- plastic composites. To evaluate the performance of transfer lea rning, the DNN model trained via transfer learning using both homogenization and FEM dataset s was compared with the DNN model trained only with a small FEM dataset. The DNN archit ectures and initial weights of the two models were identical. In addition, we used the same training, validation, and test datasets from the FE M for both DNN models. 4.1. Ellipsoidal particle reinforced composite We considered an ellipsoidal particle-reinforced composite subj ected to uniaxial tensile loading. FEM data were generated using an RVE with reinforcement particl es in the FCC configuration as shown in Fig. 2 (a, b). The prediction and ground truth of t he top three principal components are compared between Models B a nd C in Fig. 6, wh ich clearly sh ows that transfer learning is superior to direct training in p redicting principal components. The 𝑅ଶ values, which are statistical measurements representing how closely the regressio n fits the data, were much higher in Model B, as indicated in Table 2. In Model B, the 𝑅ଶ values from transfer learning were remarkably high across all 10 principal components, while in Model C, the 𝑅ଶ v a l u e s decreased significantly for less important principal components . The maximum error of each stress-strain curve is defined as follows: 15 Maximum Error ൌ max ቆ𝜎ଵ௉௥௘ௗെ𝜎ଵிாெ 𝜎ଵிாெ,…,𝜎ଵ଴ଵ௉௥௘ௗെ𝜎ଵ଴ଵிாெ 𝜎ଵ଴ଵிாெቇ. (16) The maximum error histogram is presented for 300 test data in F ig. 7. Model B, trained via transfer learning, outperformed Model C, which was directly tra ined with the FEM dataset. The predictions from both models are presented in Fig. 8 for th e four cases in which model C exhibited the greatest errors. Tr ansfer learning shows reliable predictions even for the worst cases. Moreover, Fig. S2 presents the four worst cases in which Model A, the DNN trained only with the homogenization dataset, exhibited the greatest er rors (see Supplementary Information). The comparison between the results from Models A and B clearly shows that the DNN pre-trained with the homogenization dataset is fine-tuned a ppropriately via transfer learning. 4.2. Fiber-reinforced composite Homogenization theories based on Eshelby’s solution assume that reinforcements are ellipsoidal. In this section, we verify whether transfer learni ng using homogenization data can be applied to a composite containing non-ellipsoidal reinforcem ents. The homogenization method has frequently been used to analyze composites having cy lindrical or disk-shaped reinforcements with high aspect ratios [47, 48]. In the present study, an FEA was performed on the FCC-structured RVE of a cylindrical fiber-reinforced compos ite, as shown in Fig. 2 (c, d). We generated 2,000 FEM data and di vided them into training, val idation, and test sets in a ratio of 1,400:300:300. The same homogenization dataset as that menti oned in Section 4.1 was used for transfer learning, that is, the DNN pre-trained by the homo genization dataset, where the ellipsoidal reinforcement was assumed, was fine-tuned with the FEM dataset on composites containing cylindrical fiber reinforcements. The maximum error histogram for the 300 test data is shown in Fig. 9. The predictions from both models are presen ted in Fig. 10 for the four cases 16 in which Model C exhibited the greatest errors. Fig. S3 shows t he predictions for the four cases in which Model A exhibited the greatest errors ( see supplementa ry information). As described in Section 4.1, Model B, trained via transfer learning, achieve d the best predictive performance. These results confirm that transfer learning with homogenizatio n data can also be applied to composites embedded with non-ellip soidal reinforcements. For ex ample, for composites containing flake-like reinforcements, such as graphene platelet s, a DNN can be pre-trained using homogenization data generated by assuming oblate spheroid particle reinforcements; the DNN can then be fine-tuned with FEM data or experimental data v ia transfer learning. 5. Conclusion In this study, we demonstrated a transfer learning-based compos ite analysis technique that can efficiently exploit a large dataset based on the homogenization method when only a small dataset from FEM simulations or experiments is available becaus e of high computational and economic costs. Because machine learning is a data-hungry techn ique, better generalization can be achieved with a greater amount of data. The homogenizati on theory has the advantage of providing a large amount of data at a lower cost and within a shorter time compared with the FEM or experimental methods, although its prediction accura cy is relatively inferior. Considering the task of predicti ng the elasto-plastic response of particle/fiber-reinforced composites, we demonstrated that the accuracy and efficiency of a composite analysis can be improved by training a DNN model with a large but less accurate dataset based on the homogenization method and then fine-tuning the DNN with an accu rate but small dataset based on the FEM. Although we adopted the adaptive incrementally affi ne method to generate the homogenization data, several othe r homogenization theories can also be applied to the initial training dataset in the transfer learning scheme. The proposed method can easily be generalized 17 to enable accurate and efficient prediction of various properti es of composites with only a small, accurate dataset when any of the homogenization theories can de termine the qualitative features of the composite. 18