OR3-2
Physics-informed neural network を用いたガスジェット
浮遊液滴周りの気流計算法
Prediction of flow around aerodynamically levitateddroplets based on physics-informed neural networks
吉野 裕斗 1, 白鳥英 2
Yuto YOSHINO1 and Suguru SHIRATORI2
12東京都市大学大学院，Graduate School, Tokyo City University,
東京都市大学，Tokyo City University,

1. Introduction
Sustainable human activities on the lunar surface are planned in the near future, and metal manufacturing
technology using lunar regolith will be of significant importance. Preliminary numerical simulation must be
efficient for saving resources and energy. Thus, it is essential to accurately measure the thermophysicalproperties of molten metal oxides, which are major components of lunar regolith. Although the aerodynamic
levitation (ADL) method is considered potentially applicable, its reliability as a measurement method has notbeen fully established due to the lack of a mathematical model that can quantitatively describe the effect ofsurface deformation and droplet internal flow due to the aerodynamic pressure and shear force from external
gas flow. To solve these problems, the authors’ research group is trying to develop a mathematical model fortime-averaged fields and disturbance fields deviated therefrom. The analysis of the latter oscillatory flows has
been conducted in our previous study1). This study is aiming to develop an efficient prediction procedure forthe time-averaged flow fields.
In the ADL method, the location of the droplet is determined so that the total upward aerodynamic force
is balanced with the droplet weight. The levitation force can be controlled by changing the gas flow rate. Evenfor the constant gas flow rate, the levitation force may change depending on the vertical location of droplets.
Thus, there are multiple combinations for location and flow rate that satisfy the force balance. To find suchbalanced conditions by numerical simulation, a volume-of-fluid (VOF) can be considered as an appropriate
method. However, finding many possible conditions of droplet location and gas flow rate requires huge
computational time. To solve this problem, we apply physics-informed neural networks (PINNs), as recently
proposed by Raissi et al.2). The PINNs learn the solutions of a partial differential equation (PDE) for a given
dataset. In the training process for a PINN, a loss function is defined as the mean square error of the predicted
solutions of the PDE. To evaluate the loss function, the temporal and spatial derivatives of the unknowns arecalculated by automatic differentiation (AD), which is implemented in the neural network (NN) framework.
Once a PINN has been trained, the solutions for any time instance can be calculated directly without time
integration by forward computation by the NN. In addition, the gradient of the solution with respect to theinput variable can be calculated using AD. The methodologies of the PINN are based on supervised learning;nevertheless, it does not require supervisor data, because the supervisor is assigned to the governing equation
which must equate to zero. Therefore, the PINN is expected to perform well with limited training data. The
research group of this work applied the PINNs to the liquid film flow problem. The PDE under long-wave
approximation contains 4th-order spatial derivative and 4th-order nonlinear term with respect to the Laplacepressure. Due to this term, some improvements are needed in the training of the PINNs3).

1 of 3

2. Problem formulation
Governing equations
We consider two-phase flows of incompressible Newtonian fluids of densities 𝜌𝐿 , 𝜌𝐺 and viscosities
𝜇𝐿 , 𝜇𝐺 . The subscripts 𝐿 and 𝐺 stand for liquid and gas, respectively. The surface tension of the liquid isdefined as 𝜎. The flow is governed by the conservation of mass and momentum
1 𝜕(𝑟𝑢) 𝜕𝑤+= 0,𝑟 𝜕𝑟𝜕𝑧(1a)𝜌 (𝑢𝜕𝑢𝜕𝑢𝜕𝑝𝜕 1 𝜕(𝑟𝑢)𝜕2𝑢+𝑤 )=−+𝜇[ () + 2 ] + 𝜎𝜅𝑛𝑟 𝛿𝑠 ,𝜕𝑟
𝜕𝑧𝜕𝑟
𝜕𝑟 𝑟 𝜕𝑟
𝜕𝑧(1b)𝜌 (𝑢𝜕𝑤𝜕𝑤
𝜕𝑝1𝜕𝜕𝑤
𝜕2𝑤+𝑤)=−+𝜇[(𝑟) + 2 ] + 𝜎𝜅𝑛𝑧 𝛿𝑠 − 𝜌𝑔,𝜕𝑟𝜕𝑧𝜕𝑧𝑟 𝜕𝑟 𝜕𝑟𝜕𝑧(1c)𝑢𝜕𝛼𝜕𝛼+𝑤= 0,𝜕𝑟
𝜕𝑧(1d)|∇𝜓| − 1 = 0,(1e)where 𝑢 and 𝑤 are radial and axial components of the velocity. 𝑝, 𝛼, and 𝜓 are pressure, liquid volume
fraction, and level-set function, respectively. 𝑛𝑟 , 𝑛𝑧 ,are radial and axial components of interface normal vectorand 𝜅 is a curvature of the interface. 𝛿𝑠 is the smoothed delta function, which has a nonzero value only near
the interface. 𝜌 and 𝜇 are density and viscosity of mixture phase defined as follows:𝜌 = 𝛼𝜌𝐿 + (1 − 𝛼)𝜌𝐺 ,(2)𝜇 = 𝛼𝜇𝐿 + (1 − 𝛼)𝜇𝐺 .
Physics-informed neural networks

In this study, we have been using simple deep feed-forward NN architectures, as shown in Figure 1. All
the hidden layers are fully connected dense layers, and all the activation functions are hyperbolic tangents.
The outputs of the network are selected as velocities 𝑢, 𝑤, pressure 𝑝, and level-set function 𝜓. From thepredicted 𝜓, the liquid fraction 𝛼 is calculated as1𝜓
𝛼 = [1 − tanh ( )],2𝜖(3)where 𝜖 is small constant.

utput
nput

Fig.1 Structure of physics-informed neural network used in this study.

2 of 3

Fig.2 Calculated two-phase flow fields.

The loss function for the training is defined as follows:
𝐽 = 𝐽GE + 𝐽BC + 𝐽VOL ,𝑁𝑓 𝑁𝑘

12
𝐽GE =
∑ ∑[𝐹𝑘 (𝑟𝑓𝑖 , 𝑧𝑓𝑖 )] ,𝑁𝑓𝑖=1 𝑘=1

(4)

𝑁𝑏

12𝐽BC =∑(ℬ(𝑟𝑏𝑖 , 𝑧𝑏𝑖 ) − 𝑏 𝑖 ) ,𝑁𝑏𝑖=1

where 𝐽GE is the mean squared error (MSE) of the governing equations 𝐹𝑘 (𝑟, 𝑧). The component 𝐽BC is theMSE of the boundary conditions. The component 𝐽VOL is the loss for the constraint of volume of liquid droplet.
The NN is trained to minimize the loss function 𝐽 by L-BFGS-B method, which is a quasi-Newton, full-batch
gradient-based optimization algorithm.
3. Results
Figure 2 shows the flow field predicted by the trained PINN. The color contour indicates the volume
fraction of liquid, whereas the vectors stand for the velocity. Validity and efficiency of the prediction wasconfirmed by comparison with the solution obtained by the finite volume method.
Acknowledgments
This study was partly supported by the Grant-in-Aid for Front Loading Research from the Advisory
Committee for Space Utilization Research in ISAS/JAXA.
References
1)
2)
3)

S. Shiratori, Y. Yoshino, S. Koike, S. Ozawa, Y. Seimiya, Y. Takahashi, S. Kato, H. Kobatake, K. Kawamoto, K. Sugioka,
and T. Ishikawa, 38th Space Utilization Symposium (2024).
M. Raissi, P. Perdikaris, G. Karniadakis, J. Comput. Phys. 378 (2019) 686-707, DOI: 10.1016/j.jcp.2018.10.045
Y. Nakamura, S. Shiratori, R. Takagi, M. Sutoh, I. Sugihara, H. Nagano, and K. Shimano:, Int. J. Numer. Meth.
Fluids.94 (2022) 1359-1378, DOI: 10.1002/fld.5093© 2024 by the authors. Submitted for possible open access publication under theterms and conditions of the Creative Commons Attribution (CC BY) license(http://creativecommons.org/li censes/by/4.0/).
3 of 3