discretely-trained physics-informed neural networks using meshless radial basis function-finite differences

Accelerated Training of Physics Informed Neural Networks (PINNs) using Meshless Discretizations

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-finitedifferences (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.We demonstrate the efficiency and accuracy of dt-pinns via a series of experiments.First, we explore the effect of network depth on both numericaland 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.