### 1 Introduction

### 2 Domain-adaptive Designable Data Augmentation

### 2.1 GAN with Inverse Generator

*L*) of GAN is expressed as

*D*) predicts the probability value (

*D*(

*y*)) as 1 when the actual data (

*y*) are input, and the probability value (

*D*(

*G*(

*z*))) as 0 when the virtual data (

*G*(

*z*)) generated through the latent variable (

*z*) are input. Contrastingly, generator (

*G*) aims to predict

*D*as 1 when

*D*receives the generated

*G*(

*z*). The

*G*and

*D*are trained separately. In the first step,

*y*and

*G*(

*z*) are inserted into

*D*, and training occurs. The parameters (

*θ*

*) of*

_{D}*D*are updated by increasing its stochastic gradient to maximize (1). Thus, the updated

*θ*

*can be expressed as*

_{D}*l*is the number of iterations,

*α*

*is the learning rate of*

_{D}*D*, and

*m*is the batch size of the training data. In the second step, the parameters (

*θ*

*) of*

_{G}*G*are updated by decreasing its stochastic gradient to minimize (1). Thus, the updated

*θ*

*can be expressed as*

_{G}*α*

*is the learning rate of*

_{G}*G*. Even after sufficient learning from the perspective of

*D*, if it is not possible to properly distinguish between

*y*and

*G*(

*z*), the discriminant value (or score) approaches 0.5. The value is expressed as

*G*(

*z*) are estimated using a pre-trained inverse generator. Multiple convolutions are used to downsample the input data to estimate the design variables. Before training DADDA, the parameters of the inverse generator may be determined using the actual responses (

*y*) as an input and pre-training the inverse generator (

*IG*) while using the actual design variables (

*x*) as the outputs. This can be expressed as

*θ*

*) of*

_{IG}*IG*are updated by decreasing its stochastic gradient to minimize the difference between scaled

*x*and the predicted scaled values (

*x̂*

*). Thus, the updated*

_{s}*θ*

*can be expressed as*

_{IG}*α*

*is the learning rate of*

_{IG}*IG*, and

*n*is the batch size of the pre-training data. After the

*θ*

*values are determined, DADDA is trained. For training, a pre-trained*

_{IG}*IG*is added to the GAN. The input layer of

*IG*receives

*G*(

*z*), and the output layer estimates the design variables (

*x̂*). When training the DADDA, the values for

*x̂*are outputted as predicted

*x̂*

*. This can be expressed as*

_{s}*x̂*

*output from*

_{s}*IG*was rescaled to

*x̂*according to the range of

*x*. This can be expressed as

*μ*

*and*

_{x}*σ*

*are the mean and standard deviation of*

_{x}*x*, respectively. Therefore, by adding an inverse generator to the GAN, virtual responses can be generated with only a small number of actual responses, and the design variables affecting the virtual responses can be estimated.

### 2.2 Domain Adaptation

*D*) is defined as (9), with data (

*χ*) and its probability distribution (

*P*(

*χ*))

*D*or task (

*T*) changes to a target similar to the source. Domain adaptation can be applied when the domains are different but the tasks are the same. This can be expressed as

*D*

*is the source domain,*

_{s}*D*

*is the target domain,*

_{T}*T*

*is the source task, and*

_{s}*T*

*is the target task. When the domains are different because of different data sources, they are called heterogeneous domain adaptations. When the data sources are the same but the domains are different because of the different data distributions, it is called a homogeneous domain adaptation. Homogeneous domain adaptation is frequently required in engineering design and performance prediction. Therefore, DADDA is a generative model that enables efficient design by referring to the design information required for the homogeneous domain.*

_{T}### 3 Case Study: Mathematical Example

### 3.1 Data Acquisition

*S*(

*x*

*) is the source response,*

^{s}*T*(

*x*

*) is the target response,*

^{t}*x*

*is the source design variable,*

^{s}*x*

*is the target design variable, and*

^{t}*t*is the time value. Information on

*x*

*and*

^{s}*x*

*is described in Table 1. The*

^{t}*x*

*values are known values, and the distribution of the variables is selected as a normal distribution. On the other hand, the*

^{s}*x*

*values are unknown; therefore, the distribution of the variables is selected as a uniform distribution. One hundred pieces of values were randomly extracted from the distribution of each*

^{t}*x*

*, and 20 pieces of values were randomly extracted from the distribution of each*

^{s}*x*

*. Therefore, 100 responses were acquired for the source, and 20 responses were acquired for the target, as shown in Fig. 3. The size of each response was [590 × 1 × 1], representing the data height, width, and number of channels.*

^{t}### 3.2 Pre-training of Inverse Generator

### 3.3 Training of DADDA

*f*[

*n*] is the actual curve value,

*g*[

*n*] is the virtual curve value, and

*n*is the variable of

*f*and

*g*. The similarity between

*f*[

*n*] and

*g*[

*n*] increases as WIFAC approaches 1.

### 3.4 Validation of Design Solution

### 4 Application: Generative Design of Electric Vehicle

### 4.1 Simulation-based Data Acquisition

### 4.2 Pre-training of Inverse Generator Using Simulation-based Target Dynamic Responses

### 4.3 Training of DADDA Using Simulation-based Source Dynamic Responses

### 5 Results And Discussion

### 5.1 Generation of Virtual Target Responses

*μ*

*is the average of image*

_{x}*x*;

*μ*

*is the average of image*

_{y}*y*;

*x*;

*y*;

*x*and

*y*;

*k*

_{1}is a parameter related to the image luminance, and takes 0.01;

*k*

_{2}is a parameter related to the image contrast, with a value of 0.03;

*L*is the dynamic range of the input image. If the pixel value of the data lies between 0 and 1, the dynamic range value is 1. The average accuracies of the 100 pieces of generated virtual target responses according to the number of training data are listed in Table 5. The elapsed training time increased with the number of source responses used as training data. Additionally, the accuracy of virtual target responses increased. However, it is undesirable to randomly increase the amount of data. The average accuracies derived after a certain level of increase in the amount of data were similar. The average accuracy increased from 20 to 80 data but decreased for 100 data. Therefore, the required number of source data was 80.

### 5.2 Estimation of Target Design Variables

*x*

_{2}

*represents the change in the stiffness of the rubber bush in the suspension. The*

^{t}### 5.3 Validation of Design Solution

### 6 Conclusions

1) Existing data augmentation algorithms, such as GAN, generate virtual data using a small amount of actual data. However, from the viewpoint of predicting system responses, it is difficult to determine the design variables that affect the virtual responses generated by existing data augmentation algorithms. To solve this problem, the design variables affecting the virtual responses can be estimated by adding an inverse generator to GAN. Accordingly, the performance of a system can be predicted using only a small amount of design data, that is, without conducting a large number of experiments or simulations.

2) Through domain adaptation, virtual target responses that are similar to actual source responses can be generated. Therefore, a physically reasonable performance of an actual target system can be predicted. Consequently, a new system with a similar level of performance to that of an existing system can be designed using DADDA. These deep-learning-based approaches can be effectively applied during the front-loading design stage.