Probing optimisation in physics-informed neural networks

A novel comparison is presented of the effect of optimiser choice on the accuracy of physics-informed neural networks (PINNs). To give insight into why some optimisers are better, a new approach is proposed that tracks the training trajectory curvature and can be evaluated on the fly at a low computational cost. The linear advection equation is studied for several advective velocities, and we show that the optimiser choice substantially impacts PINNs model performance and accuracy. Furthermore, using the curvature measure, we found a negative correlation between the convergence error and the curvature in the optimiser local reference frame. It is concluded that, in this case, larger local curvature values result in better solutions. Consequently, optimisation of PINNs is made more difficult as minima are in highly curved regions. The idea of solving PDE problems using neural networks (NNs) was put forward by Lagaris et al. (1997; 1998); Lagaris et al. (2000) in the second half of the '90s and then revised in 2017 by Raissi et al. (2017a;b) who named the methodology Physics-Informed Neural Networks (PINNs).