Efficient training of physics-informed neural networks via importance sampling

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 data set 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 nonconvex 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.