Hard Constraint Projection in a Physics Informed Neural Network

In this work, we embed hard constraints in a physics informed neural network (PINN) which predicts solutions to the 2D incompressible Navier Stokes equations. We extend the hard constraint method introduced by Chen et al. (arXiv:2012.06148) from a linear PDE to a strongly non-linear PDE. The PINN is used to estimate the stream function and pressure of the fluid, and by differentiating the stream function we can recover an incompressible velocity field. An unlearnable hard constraint projection (HCP) layer projects the predicted velocity and pressure to a hyperplane that admits only exact solutions to a discretised form of the governing equations.

Investigating Guiding Information for Adaptive Collocation Point Sampling in PINNs

Physics-informed neural networks (PINNs) provide a means of obtaining approximate solutions of partial differential equations and systems through the minimisation of an objective function which includes the evaluation of a residual function at a set of collocation points within the domain. The quality of a PINNs solution depends upon numerous parameters, including the number and distribution of these collocation points. In this paper we consider a number of strategies for selecting these points and investigate their impact on the overall accuracy of the method. In particular, we suggest that no single approach is likely to be ``optimal'' but we show how a number of important metrics can have an impact in improving the quality of the results obtained when using a fixed number of residual evaluations. We illustrate these approaches through the use of two benchmark test problems: Burgers' equation and the Allen-Cahn equation.