  Physics-Informed Neural Networks (PINNs) are a class of deep neural networks
that are trained, using automatic differentiation, to compute the response of
systems governed by partial differential equations (PDEs). The training of
PINNs is simulation-free, and does not require any training dataset to be
obtained from numerical PDE solvers. Instead, it only requires the physical
problem description, including the governing laws of physics, domain geometry,
initial/boundary conditions, and the material properties. This training usually
involves solving a non-convex optimization problem using variants of the
stochastic gradient descent method, with the gradient of the loss function
approximated on a batch of collocation points, selected randomly in each
iteration according to a uniform distribution. Despite the success of PINNs in
accurately solving a wide variety of PDEs, the method still requires
improvements in terms of computational efficiency. To this end, in this paper,
we study the performance of an importance sampling approach for efficient
training of PINNs. Using numerical examples together with theoretical
evidences, we show that in each training iteration, sampling the collocation
points according to a distribution proportional to the loss function will
improve the convergence behavior of the PINNs training. Additionally, we show
that providing a piecewise constant approximation to the loss function for
faster importance sampling can further improve the training efficiency. This
importance sampling approach is straightforward and easy to implement in the
existing PINN codes, and also does not introduce any new hyperparameter to
calibrate. The numerical examples include elasticity, diffusion and plane
stress problems, through which we numerically verify the accuracy and
efficiency of the importance sampling approach compared to the predominant
uniform sampling approach.
