Bound-preserving physics-informed neural networks for steady-state convection-diffusion-reaction problems

Numerical approximations of solutions of convection-diffusion-reaction problems should take only physically admissible values. Provided that bounds for the admissible values are known, this paper presents several approaches within physics-informed neural networks (PINNs) and hp-variational PINNs (hp-VPINNs) to preserve these bounds for convection-dominated problems. These approaches comprise the inclusion of the requirement for bound preservation in the cost functional, a simple cut-off strategy for the unphysical values, and two methods that enforce bound preservation via the activation function of the output layer of the neural network. Numerical simulations are performed for convection-dominated problems defined in two-dimensional domains. A variety of choices for several hyperparameters is explored. Enforcing bound preservation with the sine activation function in the output layer turned out to be superior to all other methods with respect to the accuracy of the computed solutions, and in particular, the results are much more accurate than those obtained with the standard PINNs and hp-VPINNs.