  We present a new technique for the accelerated training of physics-informed
neural networks (PINNs): discretely-trained PINNs (DT-PINNs). The repeated
computation of partial derivative terms in the PINN loss functions via
automatic differentiation during training is known to be computationally
expensive, especially for higher-order derivatives. DT-PINNs are trained by
replacing these exact spatial derivatives with high-order accurate numerical
discretizations computed using meshless radial basis function-finite
differences (RBF-FD) and applied via sparse-matrix vector multiplication. The
use of RBF-FD allows for DT-PINNs to be trained even on point cloud samples
placed on irregular domain geometries. Additionally, though traditional PINNs
(vanilla-PINNs) are typically stored and trained in 32-bit floating-point
(fp32) on the GPU, we show that for DT-PINNs, using fp64 on the GPU leads to
significantly faster training times than fp32 vanilla-PINNs with comparable
accuracy. We demonstrate the efficiency and accuracy of DT-PINNs via a series
of experiments. First, we explore the effect of network depth on both numerical
and automatic differentiation of a neural network with random weights and show
that RBF-FD approximations of third-order accuracy and above are more efficient
while being sufficiently accurate. We then compare the DT-PINNs to
vanilla-PINNs on both linear and nonlinear Poisson equations and show that
DT-PINNs achieve similar losses with 2-4x faster training times on a consumer
GPU. Finally, we also demonstrate that similar results can be obtained for the
PINN solution to the heat equation (a space-time problem) by discretizing the
spatial derivatives using RBF-FD and using automatic differentiation for the
temporal derivative. Our results show that fp64 DT-PINNs offer a superior
cost-accuracy profile to fp32 vanilla-PINNs.
