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 physics-informed 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 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. The results demonstrate the flexibility and efficiency of the proposed method, especially for inverse problems.

翻译：分数阶偏微分方程（FPDEs）能够有效表征反常输运与非局部相互作用。然而，在实际应用中，由于随机强迫或未知材料属性，天然存在固有的不确定性。考虑非局部相互作用并包含不确定性量化的数学模型可表述为随机分数阶偏微分方程（SFPDEs）。数值求解SFPDEs面临诸多挑战，尤其是长时间积分问题，因为此类问题具有高维性和非局部性。本文结合用于表征随机过程的双正交（BO）方法与用于求解偏微分方程的物理信息神经网络（PINNs），提出了双正交物理信息神经网络方法（BO-fPINN）以求解含时SFPDEs。具体而言，我们为含时SFPDEs的随机解引入深度神经网络，并按照弱形式将BO约束纳入损失函数。由于自动微分目前不适用于分数阶导数，我们采用网格离散化来计算神经网络输出的分数阶导数。BO-fPINN方法的弱形式损失函数能够克服BO方法的部分缺陷，因此可用于求解存在特征值交叉的SFPDEs。此外，BO-fPINN方法能以与正问题相同的框架和计算复杂度求解逆SFPDEs。我们通过不同基准问题验证了BO-fPINN方法的有效性，结果表明该方法具有灵活性和高效性，尤其适用于逆问题。