  Physics-Informed Neural Networks (PINNs), which incorporate PDEs as soft
constraints, train with a composite loss function that contains multiple
training point types: different types of collocation points chosen during
training to enforce each PDE and initial/boundary conditions, and experimental
points which are usually costly to obtain via experiments or simulations.
Training PINNs using this loss function is challenging as it typically requires
selecting large numbers of points of different types, each with different
training dynamics. Unlike past works that focused on the selection of either
collocation or experimental points, this work introduces PINN Adaptive
ColLocation and Experimental points selection (PINNACLE), the first algorithm
that jointly optimizes the selection of all training point types, while
automatically adjusting the proportion of collocation point types as training
progresses. PINNACLE uses information on the interaction among training point
types, which had not been considered before, based on an analysis of PINN
training dynamics via the Neural Tangent Kernel (NTK). We theoretically show
that the criterion used by PINNACLE is related to the PINN generalization
error, and empirically demonstrate that PINNACLE is able to outperform existing
point selection methods for forward, inverse, and transfer learning problems.
