A PINN-based level-set formulation for reconstruction of bubble dynamics

Solving problems in fluid mechanics plays an essential role in science and engineering, especially when it comes to optimal design, reconstruction of biomedical and geophysical flows, parameter estimation, and more. In some of these problems, only part of the data (or parameters) are known, and they fall within the broad categories of inverse and mixed problems. Thus, solving them using traditional methods is challenging or sometimes even impossible. Moreover, generating simulated data for such problems can become very costly since simulations need to be performed several times to either discover missing physics or calibrate the free parameters in the model. One possible alternative for overcoming these drawbacks is through the use of Physics-Informed Neural Networks-PINNs, in which we approximate the problem's solution using neural networks (NNs) while incorporating the known data and physical laws when training it and also easily enabling us to take advantage of computational resources like GPUs. Here, we show a Level-Set PINN-based framework for reconstructing the velocity field for bubble flows. Given only the bubble position, we apply the framework to reconstruct gas bubbles rising in viscous liquid problems. We use synthetic data generated by adaptive mesh refinement and coarsening simulations with a different method, a phase-field approach. The only data provided is a set of snapshots containing the bubble position, i.e., the phase field, from which we try to infer the velocities. Our approach does not require any reinitialization scheme, as is usual when using a level-set approach and traditional numerical methods. Such a scheme can reconstruct the flow quantities with reasonable accuracy, and it is straightforward to parallelize when using a data-parallel approach.