Anisotropic Permeability Tensor Prediction from Porous Media Microstructure via Physics-Informed Progressive Transfer Learning with Hybrid CNN-Transformer

Abstract

Accurate prediction of permeability tensors from pore-scale microstructure images is essential for subsurface flow modeling, yet direct numerical simulation requires hours per sample, fundamentally limiting large-scale uncertainty quantification and reservoir optimization workflows. A physics-informed deep learning framework is presented that resolves this bottleneck by combining a MaxViT hybrid CNN-Transformer architecture with progressive transfer learning and differentiable physical constraints. MaxViT's multi-axis attention mechanism simultaneously resolves grain-scale pore-throat geometry via block-local operations and REV-scale connectivity statistics through grid-global operations, providing the spatial hierarchy that permeability tensor prediction physically requires. Training on 20000 synthetic porous media samples spanning three orders of magnitude in permeability, a three-phase progressive curriculum advances from an ImageNet-pretrained baseline with D4-equivariant augmentation and tensor transformation, through component-weighted loss prioritizing off-diagonal coupling, to frozen-backbone transfer learning with porosity conditioning via Feature-wise Linear Modulation (FiLM). Onsager reciprocity and positive definiteness are enforced via differentiable penalty terms. On a held-out test set of 4000 samples, the framework achieves variance-weighted R2 = 0.9960 (R2_Kxx = 0.9967, R2_Kxy = 0.9758), a 33% reduction in unexplained variance over the supervised baseline. The results offer three transferable principles for physics-informed scientific machine learning: large-scale visual pretraining transfers effectively across domain boundaries; physical constraints are most robustly integrated as differentiable architectural components; and progressive training guided by diagnostic failure-mode analysis enables unambiguous attribution of performance gains across methodological stages.

Figures (12)

• Figure 1: Overview of the physics-informed progressive transfer learning pipeline for permeability tensor prediction. Input (far left): a binary porous media image $\mathbf{I}\in\{0,1\}^{128\times128}$ with porosity $\phi = 1 - \overline{\mathbf{I}}$, where pixel values of 1 denote solid grains and 0 denote pore space. Phase 2 (blue): the ImageNet-pretrained MaxViT-Base backbone processes the image through four hierarchical stages (S1--S4, channels 64--512) to produce a 768-dimensional feature vector; D4-equivariant augmentation applies all eight dihedral symmetries (three rotations shown) with consistent tensor transformation, and the physics-aware loss enforces tensor symmetry ($\lambda_{\mathrm{sym}}=0.1$). Phases are connected by checkpoint transfer arrows labeled $\theta_{\mathrm{best}}^{(k)}$. Phase 3 (green): three supplementary augmentation classes---morphological operations ($p=0.08$), elastic deformation ($p=0.08$), and cutout masking ($p=0.10$)---are introduced alongside doubled symmetry weighting ($\lambda_{\mathrm{sym}}=0.2$) and off-diagonal loss prioritization ($w_{\mathrm{off}}=1.5\,w_{\mathrm{diag}}$). Phase 4 (purple): the backbone is frozen ($\nabla_{\theta_{\mathrm{bb}}}=0$, indicated by hatching) while a porosity MLP encoder drives FiLM conditioning ($\tilde{\mathbf{x}} = \gamma_\phi \odot \mathbf{x} + \beta_\phi$) at Stages 2--4; Stochastic Weight Averaging (SWA) and Exponential Moving Average (EMA) produce three evaluation candidates from a single training run. Output (far right): the predicted permeability tensor $\hat{\mathbf{K}}\in\mathbb{R}^{2\times2}$, with representative parity plots shown for all four components ($K_{xx}$, $K_{xy}$, $K_{yx}$, $K_{yy}$).
• Figure 2: Representative binary porous media microstructures spanning the porosity and anisotropy spectrum. Each panel shows a $128\times128$ binary image (white: solid matrix; black: void space) with corresponding porosity $\phi$ and permeability tensor components. The progression from low to high porosity illustrates the transition from tortuous isolated pore channels (small diagonal permeability, negligible off-diagonal coupling) to well-connected isotropic networks (large nearly equal $K_{xx}$, $K_{yy}$). Samples with significant off-diagonal components ($|K_{xy}|>0.05$) are associated with oblique pore-channel orientations rather than porosity magnitude, consistent with the near-zero porosity-to-off-diagonal correlation ($\rho_s\approx0.01$) reported in Appendix \ref{['app:phi_perm']}.
• Figure 3: Component-wise porosity-permeability relationships across all 20,000 labeled samples, shown as hexbin density plots with marginal histograms. Color intensity indicates sample density. (a) $K_{xx}$ versus $\phi$: strong positive correlation ($\rho_s=+0.940$, $r=+0.795$), spanning three orders of magnitude; scatter increases at high porosity, reflecting geometric diversity beyond scalar pore fraction. (b) $K_{yy}$ versus $\phi$: nearly identical statistics and correlation structure to $K_{xx}$, confirming statistical isotropy at the dataset level despite individual-sample anisotropy. (c) $K_{xy}$ versus $\phi$: negligible correlation ($\rho_s\approx+0.01$, $p>0.05$); off-diagonal coupling clusters symmetrically near zero, independent of pore volume fraction. (d) $K_{yx}$ versus $\phi$: identical to $K_{xy}$, confirming near-machine-precision ground-truth symmetry ($\varepsilon_{\mathrm{sym}}<8\times10^{-6}$, Table \ref{['tab:dataset_stats']}). The contrast between panels (a,b) and (c,d) constitutes the fundamental prediction-difficulty asymmetry between diagonal and off-diagonal tensor components that the progressive training strategy is designed to address.
• Figure 4: Component-wise parity plots for Phase 2 permeability tensor predictions. Each panel shows predicted versus true values for one tensor component, with hexbin color intensity proportional to log-density. Marginal histograms compare the distributions of true (gray) and predicted (colored) values. (a) $K_{xx}$ ($R^2=0.9961$, RRMSE $=5.77$ %): predictions closely track the identity line across the full three-order-of-magnitude range; slight scatter at extreme values reflects the sparsely populated tails of the training distribution. (b) $K_{yy}$ ($R^2=0.9962$, RRMSE $=5.70$ %): nearly identical accuracy to $K_{xx}$, confirming statistical isotropy in the backbone's ability to resolve principal permeabilities. (c) $K_{xy}$ ($R^2=0.9725$, RRMSE $=26.07$ %): increased scatter around the identity line at large $|K_{xy}|$ values, with the marginal histogram revealing systematic under-dispersion (KGE $=0.176$). (d) $K_{yx}$ ($R^2=0.9725$, RRMSE $=26.07$ %): essentially identical to $K_{xy}$, confirming that the symmetry constraint $K_{xy}\approx K_{yx}$ is learned accurately ($\varepsilon_{\mathrm{sym}}=1.39\times10^{-6}$, near-machine precision). The 236-basis-point $R^2$ gap between diagonal and off-diagonal components drives all subsequent Phase 3 interventions.
• Figure 5: Anisotropy-stratified prediction performance for Phase 2 across ten equi-populated bins ($n=400$ each) of tensor anisotropy ratio $\mathrm{AR}=\lambda_{\max}/\lambda_{\min}$. (a) AR distribution of the 4,000 test samples (log-scale $x$-axis); vertical dashed lines mark the 50th, 90th, and 99th percentiles ($\mathrm{AR}_{50}=1.62$, $\mathrm{AR}_{90}=3.34$, $\mathrm{AR}_{99}=10.0$). (b) $R^2$ for diagonal (mean of $K_{xx}$, $K_{yy}$) and off-diagonal (mean of $K_{xy}$, $K_{yx}$) components versus AR bin center: the diagonal-to-off-diagonal gap is largest at the lowest anisotropy ratios ($\Delta R^2=0.194$ at AR $\approx1.11$), narrows rapidly with increasing AR ($\Delta R^2<0.02$ for AR $>1.5$), and exhibits a modest secondary widening at high AR ($\Delta R^2=0.027$ at AR $\approx5.87$). This non-monotonic pattern identifies weak off-diagonal coupling in near-isotropic media as the dominant Phase 2 failure mode. (c) RMSE versus AR bin center for all four components; off-diagonal RMSE is elevated at both extremes of the AR spectrum while diagonal RMSE remains largely insensitive throughout. (d) Residual distributions per component (violin plots); off-diagonal interquartile ranges are widest at low AR, where near-zero coupling values produce low-variance targets that amplify the relative impact of prediction errors.