Physics-Informed PointNet: A Deep Learning Strategy for Solving Nonlinear PDEs on Unseen Irregular Geometries

Physics Informed Neural Network (PINN) is a semi-supervised deep learning methodology for solving partial differential equations (PDEs) governing physical phenomena. In PINNs, instead of using labeled data for training, the physics is incorporated in the loss function as the mean squared residuals of the governing PDEs and the associated boundary conditions. However, current versions of PINNs are restricted to a fixed geometry that was used in the training, requiring retraining of the network for any new geometry. Thus, the goal of using deep learning for accelerating geometric optimization in industrial designs is currently unreachable through PINNs. To obviate this limitation, we introduce a Physics Informed PointNet (PIPN) framework. PIPN, once trained, can predict the solution of PDEs of interest not only without labeled data but also on an unseen set of irregular geometries. As a test case, we consider PDEs of conservation of mass, momentum, and energy for incompressible flow with two specific examples: the method of manufactured solution in nontrivial geometries; and natural convection in a square enclosure with a cylinder with various shapes for its cross section. A study of prediction accuracy and obtained speedup factor compared to regular CFD solvers is presented.