Monte Carlo PINNs: deep learning approach for forward and inverse problems involving high dimensional fractional partial differential equations

Understand

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.