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Physics-informed neural networks (PINNs) incorporate physical knowledge fromthe problem domain as a soft constraint on the loss function, but recent workhas shown that this can lead to optimization difficulties. Here, we study theimpact of the location of the collocation points on the trainability of thesemodels. We find that the vanilla PINN performance can be significantly boostedby adapting the location of the collocation points as training proceeds.Specifically, we propose a novel adaptive collocation scheme whichprogressively allocates more collocation points (without increasing theirnumber) to areas where the model is making higher errors (based on the gradientof the loss function in the domain). This, coupled with a judicious restartingof the training during any optimization stalls (by simply resampling thecollocation points in order to adjust the loss landscape) leads to betterestimates for the prediction error. We present results for several problems,including a 2D Poisson and diffusion-advection system with different forcingfunctions. We find that training vanilla PINNs for these problems can result inup to 70% prediction error in the solution, especially in the regime of lowcollocation points. In contrast, our adaptive schemes can achieve up to anorder of magnitude smaller error, with similar computational complexity as thebaseline. Furthermore, we find that the adaptive methods consistently performon-par or slightly better than vanilla PINN method, even for large collocationpoint regimes. The code for all the experiments has been open sourced.

Adaptive Self-supervision Algorithms for Physics-informed Neural Networks