Bi-Orthogonal fPINN:A Physics-Informed Neural Network Method for Solving Time-Dependent Stochastic Fractional PDEs

Fractional partial differential equations(FPDEs)can effectively represent anomalous transport and nonlocal interactions.However,inherent uncertainties arise naturally in real applications due to random forcing or unknown material properties.Mathematical models considering nonlocal interactions with uncertainty quantification can be formulated as stochastic fractional partial differential equations(SFPDEs).There are many challenges in solving SFPDEs numerically,especially for long-time integration since such problems are high-dimensional and nonlocal.Here,we combine the bi-orthogonal(BO)method for representing stochastic processes with physicsinformed neural networks(PINNs)for solving partial differential equations to formulate the bi-orthogonal PINN method(BO-fPINN)for solving time-dependent SFPDEs.Specifically,we introduce a deep neural network for the stochastic solution of the time-dependent SFPDEs,and include the BO constraints in the loss function following a weak formulation.Since automatic differentiation is not currently applicable to fractional derivatives,we employ discretization on a grid to compute the fractional derivatives of the neural network output.The weak formulation loss function of the BO-fPINN method can overcome some drawbacks of the BO methods and thus can be used to solve SFPDEs with eigenvalue crossings.Moreover,the BO-fPINN method can be used for inverse SFPDEs with the same framework and same computational complexity as for forward problems.We demonstrate the effectiveness of the BO-fPINN method for different benchmark problems.Specifically,we first consider an SFPDE with eigenvalue crossing and obtain good results while the original BO method fails.We then solve several forward and inverse problems governed by SFPDEs,including problems with noisy initial conditions.We study the effect of the fractional order as well as the number of the BO modes on the accuracy of the BO-fPINN method.The results demonstrate the flexibility and efficiency of the proposed method,especially for inverse problems.We also present a simple example of transfer learning(for the fractional order)that can help in accelerating the training of BO-fPINN for SFPDEs.Taken together,the simulation results show that the BO-fPINN method can be employed to effectively solve time-dependent SFPDEs and may provide a reliable computational strategy for real applications exhibiting anomalous transport.