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Register a new userPhysics-informed neural network applied to surface-tension-driven liquid film flows
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Added by
Suguru Shiratori Dr. Eng.
Publication date
2021
fields
Physics
and research's language is
English
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A physics-informed neural network (PINN), which has been recently proposed by Raissi et al [J. Comp. Phys. 378, pp. 686-707 (2019)], is applied to the partial differential equation (PDE) of liquid film flows. The PDE considered is the time evolution of the thickness distribution $h(x,t)$ owing to the Laplace pressure, which involves 4th-order spatial derivative and 4th-order nonlinear term. Even for such a PDE, it is confirmed that the PINN can predict the solutions with sufficient accuracy. Nevertheless, some improvements are needed in training convergence and accuracy of the solutions. The precision of floating-point numbers is a critical issue for the present PDE. When the calculation is executed with a single precision floating-point number, the optimization is terminated due to the loss of significant digits. Calculation of the automatic differentiation (AD) dominates the computational time required for training, and becomes exponentially longer with increasing order of derivatives. By splitting the original 4th-order one-variable PDE into 2nd-order two-variable PDEs, the computational time for each training iteration is greatly reduced. The sampling density of training data also significantly affects training convergence. For the problem considered in this study, mproved convergence was obtained by allowing the sampling density of training data to be greater in earlier time ranges, where the rapid diffusion of the thickness occurs.
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We propose a discretization-free approach based on the physics-informed neural network (PINN) method for solving coupled advection-dispersion and Darcy flow equations with space-dependent hydraulic conductivity. In this approach, the hydraulic conductivity, hydraulic head, and concentration fields are approximated with deep neural networks (DNNs). We assume that the conductivity field is given by its values on a grid, and we use these values to train the conductivity DNN. The head and concentration DNNs are trained by minimizing the residuals of the flow equation and ADE and using the initial and boundary conditions as additional constraints. The PINN method is applied to one- and two-dimensional forward advection-dispersion equations (ADEs), where its performance for various P{e}clet numbers ($Pe$) is compared with the analytical and numerical solutions. We find that the PINN method is accurate with errors of less than 1% and outperforms some conventional discretization-based methods for $Pe$ larger than 100. Next, we demonstrate that the PINN method remains accurate for the backward ADEs, with the relative errors in most cases staying under 5% compared to the reference concentration field. Finally, we show that when available, the concentration measurements can be easily incorporated in the PINN method and significantly improve (by more than 50% in the considered cases) the accuracy of the PINN solution of the backward ADE.
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It was recently claimed by Bhagat et al. (J. Fluid Mech. vol. 851 (2018), R5) that the scientific literature on the circular hydraulic jump in a thin liquid film is flawed by improper treatment and severe underestimation of the influence of surface tension. Bhagat {em et al.} use an energy equation with a new surface energy term that is introduced without reference, and they conclude that the location of the hydraulic jump is determined by surface tension alone. We show that this approach is incorrect and derive a corrected energy equation. Proper treatment of surface tension in thin film flows is of general interest beyond hydraulic jumps, and we show that the effect of surface tension is fully contained in the Laplace pressure due to the curvature of the surface. Following the same approach as Bhagat et al., i.e., keeping only the first derivative of the surface velocity, the influence of surface tension is, for thin films, much smaller than claimed by them. We further describe the influence of viscosity in thin film flows, and we conclude by discussing the distinction between time-dependent and stationary hydraulic jumps.
Recently, physics-driven deep learning methods have shown particular promise for the prediction of physical fields, especially to reduce the dependency on large amounts of pre-computed training data. In this work, we target the physics-driven learning of complex flow fields with high resolutions. We propose the use of emph{Convolutional neural networks} (CNN) based U-net architectures to efficiently represent and reconstruct the input and output fields, respectively. By introducing Navier-Stokes equations and boundary conditions into loss functions, the physics-driven CNN is designed to predict corresponding steady flow fields directly. In particular, this prevents many of the difficulties associated with approaches employing fully connected neural networks. Several numerical experiments are conducted to investigate the behavior of the CNN approach, and the results indicate that a first-order accuracy has been achieved. Specifically for the case of a flow around a cylinder, different flow regimes can be learned and the adhered twin-vortices are predicted correctly. The numerical results also show that the training for multiple cases is accelerated significantly, especially for the difficult cases at low Reynolds numbers, and when limited reference solutions are used as supplementary learning targets.
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