A Discrete Neural Operator with Adaptive Sampling for Surrogate Modeling of Parametric Transient Darcy Flows in Porous Media

TL;DR

The paper presents a discrete neural operator, AROnet, with time-embedded operation for surrogate modeling of transient Darcy flows in heterogeneous porous media. It shows that using transmissibility inputs from finite-volume discretization and a branch-trunk operator architecture yields higher accuracy than a state-of-the-art Attentional Residual U-net, especially with limited data. An adaptive sampling scheme in latent space based on Gaussian mixtures further improves predictions by targeting high-error regions while keeping computation reasonable. The approach is validated on 2D/3D single- and two-phase Darcy flows, demonstrating consistent accuracy gains and practical potential for real-time subsurface simulations.

Abstract

This study proposes a new discrete neural operator for surrogate modeling of transient Darcy flow fields in heterogeneous porous media with random parameters. The new method integrates temporal encoding, operator learning and UNet to approximate the mapping between vector spaces of random parameter and spatiotemporal flow fields. The new discrete neural operator can achieve higher prediction accuracy than the SOTA attention-residual-UNet structure. Derived from the finite volume method, the transmissibility matrices rather than permeability is adopted as the inputs of surrogates to enhance the prediction accuracy further. To increase sampling efficiency, a generative latent space adaptive sampling method is developed employing the Gaussian mixture model for density estimation of generalization error. Validation is conducted on test cases of 2D/3D single- and two-phase Darcy flow field prediction. Results reveal consistent enhancement in prediction accuracy given limited training set.

Figures (14)

• Figure 1: (a): Attentional Residual U-net structure. (b): An example computational process of ARUnet, using an image data with shape [batch size, channels, hight, weight] = [1, 1, 20, 20] as input of NN.
• Figure 2: (a): ARUnet Temporal Prediction Framework. (b): AROnet Operator Learning Framework.
• Figure 3: Adaptive Sampling in Latent Space.
• Figure 4: (a): Heterogeneous permeability field generated by sequential Gaussian simulation. (b): Corresponding pressure computed by FVM at last time step.
• Figure 5: The ARUnet surrogates are trained with the same samples but different input feature to predict the pressure field. (a) $\log_{10}{\text{relative error}}$ distribution on all grid cells in the test set. (b) $\log_{10}{\text{relative error}}$ on the well-block cell in the test set.