  We introduce a sampling based machine learning approach, Monte Carlo physics
informed neural networks (MC-PINNs), for solving forward and inverse fractional
partial differential equations (FPDEs). As a generalization of physics informed
neural networks (PINNs), our method relies on deep neural network surrogates in
addition to a stochastic approximation strategy for computing the fractional
derivatives of the DNN outputs. A key ingredient in our MC-PINNs is to
construct an unbiased estimation of the physical soft constraints in the loss
function. Our directly sampling approach can yield less overall computational
cost compared to fPINNs proposed in \cite{pang2019fpinns} and thus provide an
opportunity for solving high dimensional fractional PDEs. We validate the
performance of MC-PINNs method via several examples that include high
dimensional integral fractional Laplacian equations, parametric identification
of time-space fractional PDEs, and fractional diffusion equation with random
inputs. The results show that MC-PINNs is flexible and promising to tackle
high-dimensional FPDEs.
