GeONet: Geometry-Encoded Operator Network for Predicting Flow in Heterogeneous Porous Media

Md Lal Mamud , Maruti Mudunuru, Yucheng Fu, Aramy Troung, Satish Karra, Ivani Patel, Arunima Bhattacharjee, Tamas Varga, Pratanu Roy, Michael Penwarden, Rajesh K. Singh, Arjun Chakrawal, Kalyana B. Nakshatrala, Christof Meile, Erin Rooney, Glenn Hammond

Rapid prediction of pressure-driven flow in microfluidic porous-media analogs is important for experimental design, tracer-test interpretation, and real-time digital-twin development for subsurface energy and critical mineral science applications. High-fidelity numerical simulators resolve geometry-controlled flow in heterogeneous pore networks, but their computational cost limits repeated evaluation across large geometry ensembles. Here, we introduce GeONet, a Geometry-Encoded Operator Network that predicts steady-state pressure fields by jointly representing material properties, spatial coordinates, and explicit solid–void geometry. GeONet extends the conventional Branch–Trunk architecture with a dedicated Leaf Network that encodes the binary domain mask through multiscale features. The Branch Network represents porosity and permeability, the Trunk Network combines coordinates with local geometric descriptors and random Fourier features, and a gated DeepMixer integrates the three latent representations before decoding pressure. GeONet was trained and evaluated using 5,000 PFLOTRAN simulations of heterogeneous Reduced Complexity Platform domains, comprising nearly 30 TB of data. Direct comparisons with pore-scale Stokes solutions across three geometries yielded Pearson correlations of 0.987–0.991 and R² values of 0.974–0.982, supporting PFLOTRAN solutions as reference data. Under matched data partitions and two-stage hyperparameter optimization, GeONet substantially outperformed a boundary-condition-aware DeepONet baseline. Across three held-out test domains, GeONet achieved NRMSE of 0.0381–0.0500, MAPE of 1.76–2.59%, and R² of 0.9673–0.9822, compared with 0.1483–0.2040, 7.81–10.45%, and 0.5658–0.7126 for DeepONet, respectively. GeONet generalized to three additional unseen geometries, achieving NRMSE of 0.0448–0.0676 and R² of 0.9511–0.9770. These results demonstrate that explicit geometry encoding improves accuracy, pressure-range preservation, and cross-geometry generalization of neural operators for flow in heterogeneous porous domains.