Power series physics-informed neural networks for variable-order fractional diffusion: Forward modeling, joint inversion, and uncertainty quantification in heterogeneous porous media

Variable-order fractional partial differential equations describe anomalous transport in heterogeneous porous media, where the fractional order encodes the spatial variability of the medium. In subsurface applications such as groundwater flow and oil reservoir characterization, the fractional order is typically unknown and varies spatially, making its recovery alongside the concentration field an ill-posed inverse problem. We propose a physics-informed neural network (PINN) that jointly recovers the space-dependent fractional order $ \alpha \left(x\right) $ and the concentration field $ u(x, t) $ from sparse, noise-contaminated measurements. The method represents the solution through a power series expansion in space, which permits analytical computation of fractional derivatives via Gamma function identities and circumvents the need for numerical discretization. A dual-network architecture separates solution reconstruction from parameter inference: One network learns time-dependent series coefficients, while the other estimates the fractional order. The framework is validated on four geological configurations, fractured media, fault zones, dual-porosity systems, and field-like heterogeneous formations, using synthetic monitoring well data with realistic noise levels.

For the forward problem, the power series PINN achieves excellent accuracy with $ L2 $ relative errors on the order of $ {10}^{-4} $ when using $ N = 12 $ expansion terms. The joint inversion problem poses greater challenges: With the current configuration using $ N = 8 $ and $ \mathrm{10,121} $ parameters, the $ L2 $ relative error in the concentration field is $ 3.93\times {10}^{-1} $ and the MSE in $ \alpha \left(x\right) $ is $ 2.92\times {10}^{-1} $. These results highlight the ill-posed nature of recovering both the heterogeneity parameter and the concentration field from sparse data.

In the real-world application to the fractured media scenario with $ N = 12 $, the framework recovers $ \alpha \left(x\right) $ with an MSE of $ 4.56\times {10}^{-5} $ and reconstructs the concentration field with $ L2 $ relative error of 2.34 × 10^-3 using eight monitoring wells with 1.5% observation noise. For uncertainty quantification, we implement both Monte Carlo Dropout and ensemble-based techniques. The ensemble method yields $ 95\% $ confidence intervals for $ \alpha \left(x\right) $ with coverage rates of $ 94.2\% $, surpassing MC Dropout at $ 92.1\% $ coverage. The ensemble approach also pinpoints spatial regions where the recovered heterogeneity is most uncertain, offering practical guidance for optimizing monitoring network design. The overall training requires roughly two hours on a standard GPU and demonstrates consistent performance across diverse heterogeneity patterns. The proposed framework provides a computationally feasible and uncertainty-aware tool for subsurface characterization that bridges fractional calculus models with practical data constraints.