# A Short Note on Physics-Guided GAN to Learn Physical Models without Gradients
## 1 Introduction  
Deep neural networks (DNNs) are widely employed in diverse research areas, including physics and engineering. In mechanical engineering, DNNs are often used as a surrogate model [1; 2; 3]. In general, a DNN learns from data, and the output is not guaranteed to be consistent with physics, even if the data were generated by certain physical models. Generative adversarial networks (GANs) [4] constitute a type of DNN. GANs are applied to image generation tasks at first [5; 6; 7]. Then, GANs are also used to solve inverse design problems [8; 9; 10; 11]. For example, in [12; 13], the GAN models were trained using a set of data consisting of airfoil shapes and their aerodynamic coefficients. Then, by inputting the aerodynamic coefficients, the trained model output airfoil shapes associated with those inputted aerodynamic coefficients. The aerodynamic coefficients of the generated shapes were close to the specified label, but some errors were also identified. These errors were due to the fact that the aerodynamic coefficient was calculated from the airfoil shapes, but the physical equations were not inputted into the DNN model. To consider physical consistency, physics-informed neural networks (PINNs) have been proposed [14], which add the residual of physical equations to the loss function of the DNN [2; 15; 16; 17]. PINNs were used to predict various targets, such as lake water levels [18], surface water levels [19], and seismic responses [20]. However, a PINN model needs the gradients of the physical equations, and hence the physical equations must be implemented in the DNN architecture. This causes difficulties from an application point of view. For example, general purpose software and commercial software cannot be used in PINN models. Especially in the inverse design problems, the commercial software are often mandatory to compute the label; e.g., flow computations to calculate aerodynamic coefficients in the airfoil shapes are mandatory. Hence, it is desirable to consider arbitraryphysical equations, including general purpose and commercial software in DNN surrogate models and generative models. The proposed method aims to consider arbitrary physical equations using a GAN architecture. To this aim, the gradients of the physical equations have to be eliminated from the algorithm. GAN consists of a generator network and a discriminator network. The generator network outputs data that mimic training data, whereas the discriminator network distinguishes the generated data from the training data. Training data are referred to as true data, whereas generated data constitute fake data. In the literature, physics-informed adversarial learning [16; 21] and PID-GAN [22], which couples PINN and GAN, are proposed, but those methods also use the residual of the physical equations in the same manner as PINNs, and hence the physical models need to be implemented inside the computation graph. In the proposed PG-GAN model, true or fake is defined by physical equations; if the residual is smaller than a specific value, then the generated data are true; otherwise, the generated data are fake. The physical model guides the DNN to learn physical consistency, and is only used to categorize data as true or fake. The physical equations remain outside of the DNN model, and are not implemented inside the DNN. Therefore, arbitrary physical models can be used. It is also noted that, by decreasing the threshold value, the residual of the generated data are decreased. In the PINN model, the residual is added to the loss function and cannot be controlled. The concept of PG-GAN model is validated using a simple example. The merit of the PG-GAN is that an arbitrary physics model can be used. However, to compare with PINN, [PERSON]'s equation of motion is employed, which can be solved by PINN. The present paper is organized as follows. GAN and PINN models are explained in Section 2. The concept of PG-GAN is explained in Section 3, and its formulation is given in Section 3. A numerical study is presented in Section 4.