  Physics-informed neural networks (PINNs) are effective in solving
integer-order partial differential equations (PDEs) based on scattered and
noisy data. PINNs employ standard feedforward neural networks (NNs) with the
PDEs explicitly encoded into the NN using automatic differentiation, while the
sum of the mean-squared PDE-residuals and the mean-squared error in
initial/boundary conditions is minimized with respect to the NN parameters. We
extend PINNs to fractional PINNs (fPINNs) to solve space-time fractional
advection-diffusion equations (fractional ADEs), and we demonstrate their
accuracy and effectiveness in solving multi-dimensional forward and inverse
problems with forcing terms whose values are only known at randomly scattered
spatio-temporal coordinates (black-box forcing terms). A novel element of the
fPINNs is the hybrid approach that we introduce for constructing the residual
in the loss function using both automatic differentiation for the integer-order
operators and numerical discretization for the fractional operators. We
consider 1D time-dependent fractional ADEs and compare white-box (WB) and
black-box (BB) forcing. We observe that for the BB forcing fPINNs outperform
FDM. Subsequently, we consider multi-dimensional time-, space-, and
space-time-fractional ADEs using the directional fractional Laplacian and we
observe relative errors of $10^{-4}$. Finally, we solve several inverse
problems in 1D, 2D, and 3D to identify the fractional orders, diffusion
coefficients, and transport velocities and obtain accurate results even in the
presence of significant noise.

Computational Physics