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

Mohammad Nooraiepour

cs.LGMar 18, 2026

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

Authors: Mohammad Nooraiepour

Organizations: Faculty of Mathematics and Natural Sciences, University of Oslo, PO Box 1047 Blindern, 0316 Oslo, Norway

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 & tables

Figure 1: Overview of the physics-informed progressive transfer learning pipeline for permeability tensor prediction. Input (far left): a binary porous media image $\mathbf{I}\in\{0,1\}^{128\times 128}$ 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 $\hat{\mathbf{K}}\in\mathbb{R}^{2\times 2}$ , 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 $128\times 128$ 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 ( $\rho_{s}\approx 0.01$ ) reported in Appendix A.3 .

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 ( $\varepsilon_{\mathrm{sym}}<8\times 10^{-6}$ , Table 3 ). 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.

Component	$R^{2}$	RMSE	MAE	RRMSE (%)	KGE	$d$
$K_{xx}$	0.9961	$2.147\times 10^{-2}$	$1.430\times 10^{-2}$	5.77	0.986	0.9990
$K_{yy}$	0.9962	$2.124\times 10^{-2}$	$1.394\times 10^{-2}$	5.70	0.989	0.9990
$K_{xy}$	0.9725	$1.095\times 10^{-2}$	$7.70\times 10^{-3}$	26.07	0.176	0.9930
$K_{yx}$	0.9725	$1.095\times 10^{-2}$	$7.70\times 10^{-3}$	26.07	0.176	0.9930
Overall	0.9843	$\mathbf{1.796\times 10^{-2}}$	$\mathbf{1.091\times 10^{-2}}$	8.19	0.988	0.9992

Table 1: Phase 2 test-set performance by tensor component. RRMSE: relative root mean squared error (normalized by component mean absolute value; Equation 23 ). MAPE values for off-diagonal components are large because the denominator (mean absolute true value) approaches zero; RRMSE is the appropriate relative-error metric for near-zero-mean components. KGE: Kling-Gupta efficiency [ 32 ] .

d

: Willmott index of agreement [ 78 ] .

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 ( $\varepsilon_{\mathrm{sym}}=1.39\times 10^{-6}$ , 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

\approx 1.11

), 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

\approx 5.87

). 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.

Metric		Phase 4	Phase 3	Phase 2
Global accuracy
	$R^{2}$ (variance-weighted)	0.9960	0.9945	0.9940
	$R^{2}$ (component-averaged)	0.9863	0.9848	0.9843
	95 % BCa CI	[0.9852, 0.9874]	[0.9833, 0.9860]	[0.9830, 0.9856]
	RMSE	$\mathbf{1.564\times 10^{-2}}$	$1.697\times 10^{-2}$	$1.796\times 10^{-2}$
	95 % BCa CI	[0.01483, 0.01704]	[0.01617, 0.01794]	[0.01694, 0.01948]

Table 2: Progressive performance across training phases on the held-out test set (

N=4{,}000

). 95 % BCa bootstrap confidence intervals (

B=1{,}000

) in brackets. Bold values indicate the best result per row. Phase 2: supervised baseline with D4 augmentation; Phase 3: advanced augmentation and enhanced physics-aware loss; Phase 4: frozen backbone, FiLM porosity conditioning, SWA/EMA ensemble.

Figure 6: Component-wise parity plots for Phase 4 (hexbin density, marginal histograms). Color encodes log-density; marginals compare true (gray) and predicted (colored) distributions. (a) $K_{xx}$ : $R^{2}=0.9967$ , RRMSE $=5.25$ %, absolute bias $<1.3\times 10^{-4}$ ; predictions track the identity across three orders of magnitude without systematic deviation. (b) $K_{yy}$ : $R^{2}=0.9967$ , RRMSE $=5.28$ %; accuracy is symmetric between the two principal permeabilities, confirming that the MaxViT backbone resolves both flow directions with equal fidelity. (c) $K_{xy}$ : $R^{2}=0.9758$ , RRMSE $=24.43$ %; the predicted marginal distribution closely matches the ground truth, including the near-zero peak and heavy tails; KGE $=0.761$ , a 4.3 $\times$ improvement over Phase 2 ( $0.176$ ), marks the near-elimination of variability under-dispersion. (d) $K_{yx}$ : statistics identical to $K_{xy}$ ; symmetry error $\varepsilon_{\mathrm{sym}}=3.95\times 10^{-7}$ (95 % BCa CI: $[3.82,\,4.10]\times 10^{-7}$ ) confirms near-machine-precision Onsager reciprocity across the full off-diagonal magnitude range.

Figure 7: Phase 4 predictions across the porosity spectrum. Each panel shows the binary microstructure ( $128\times 128$ ; white: solid grain, black: pore void), ground-truth LBM permeability tensor, and Phase 4 predicted tensor. At high porosity ( $\phi>0.70$ ), well-connected pore networks yield smooth structure-property mappings; Frobenius errors fall below 5 % and predicted condition numbers match ground truth to within 0.15. At moderate porosity ( $0.50\leq\phi\leq 0.70$ ), tensor magnitude, principal-axis orientation, and off-diagonal coupling sign and magnitude are recovered accurately across geometrically diverse configurations, including cases with condition numbers exceeding 10. At low porosity ( $\phi<0.50$ ), predictions correctly identify anisotropy direction and off-diagonal coupling sign; absolute magnitude uncertainty increases toward the lower tail of the training distribution (Appendix A.3 ). All displayed tensors are positive-definite with $\varepsilon_{\mathrm{sym}}<6\times 10^{-6}$ .

Figure 8: Anisotropy-stratified representative predictions for Phase 4. Ten samples selected from log-spaced AR bins ( $\mathrm{AR}=1.11$ to $5.87$ ) span the 5th–95th percentile of the test AR distribution. Each panel displays the binary microstructure, AR, porosity $\phi$ , ground-truth tensor, and Phase 4 prediction. At near-isotropic AR ( $\approx 1.11$ ), off-diagonal coupling is small and residual prediction uncertainty is concentrated in magnitude rather than direction—consistent with the variance-normalization effect in Section 3.2.2 ; $\Delta R^{2}$ at this bin is 0.181, down from 0.194 in Phase 2. At moderate AR ( $1.5$ – $3.0$ ), coupling magnitude and tensor orientation are resolved accurately, with $\Delta R^{2}<0.015$ . At high AR ( $\approx 5.87$ ), the dominant flow axis is correctly identified; off-diagonal $R^{2}=0.970$ in this bin. Full stratified metrics for all ten AR bins are in Appendix I .

Figure 9: Learned porosity-permeability relationship for Phase 4 ( $N=4{,}000$ test samples). (a) $K_{xx}$ versus $\phi$ on logarithmic scale: predicted and true values show nearly identical Pearson correlations ( $r_{\mathrm{pred}}=0.799$ , $r_{\mathrm{true}}=0.795$ , both $p<10^{-15}$ ); order-of-magnitude scatter at fixed $\phi$ confirms that porosity alone is insufficient for accurate prediction. (b) Permeability trace ( $K_{xx}+K_{yy}$ ) versus $\phi$ : higher correlation ( $r=0.826$ , $p<10^{-15}$ ), confirming that total transport capacity scaling with pore volume fraction is well captured. (c) Binned mean $K_{xx}$ across ten equally spaced $\phi$ bins: predicted means track ground truth with mean absolute deviation $<0.005$ in 9 of 10 bins, validating continuous structure-property learning without regime-specific discontinuities. (d) Absolute prediction error for $K_{xx}$ versus $\phi$ : positive correlation ( $r=0.343$ , slope $0.041$ ) reflects heteroscedastic absolute errors; relative errors are homoscedastic or improve with porosity (Q1: 14 %, Q4: 4 %), confirming that larger high- $\phi$ errors reflect dynamic range, not degraded accuracy.

Figure 10: Physical constraint validation for Phase 4 ( $N=4{,}000$ test tensors). (a) Predicted $K_{xy}$ versus $K_{yx}$ ; linear regression yields slope $0.999996$ , intercept $1.16\times 10^{-7}$ , confirming near-perfect Onsager reciprocity. (b) Symmetry error distribution (log scale): mean $\varepsilon_{\mathrm{sym}}=3.95\times 10^{-7}$ ; 95 % of predictions below $1.14\times 10^{-6}$ ; maximum violation $5.60\times 10^{-6}$ ; zero samples exceed $10^{-5}$ . (c) Minimum ( $\lambda_{\min}$ ) and maximum ( $\lambda_{\max}$ ) eigenvalue distributions: $\lambda_{\min}$ spans $[2.27\times 10^{-3},\,1.51]$ (mean $0.295$ ); 100 % of samples satisfy $\lambda_{\min}>0$ . (d) Condition number distribution: median $\kappa=1.59$ , 99.72 % well-conditioned ( $\kappa<10$ ), maximum $\kappa=23.5$ ; zero pathologically conditioned tensors.

Figure 11: MC-Dropout predictive uncertainty for Phase 4 ( $T=100$ passes, $N=4{,}000$ test samples). (a) Calibration curve: nominal versus empirical coverage; MCE $=0.3724$ , improving monotonically from Phase 2 ( $0.3786$ ) through Phase 4. (b) Per-component uncertainty distributions (CCDF); diagonal median uncertainty $\tilde{\sigma}\approx 9.1\times 10^{-3}$ ; off-diagonal $\tilde{\sigma}\approx 3.7\times 10^{-3}$ , proportionate to component dynamic range. (c) MC-Dropout standard deviation versus absolute prediction error; Spearman $\rho_{s}(\sigma,|\varepsilon|)=0.229$ (diagonal) to $0.304$ (off-diagonal), confirming statistically meaningful error-ranking capability. (d) Spatial uncertainty maps for representative test samples.

Figure 12: Bootstrap statistical portrait and anisotropy-stratified performance across phases. (a) Relative 95 % BCa CI width (tornado chart):

R^{2}

CIs span

<0.0022

, RRMSE CIs span

<1.01

%, symmetry error CI spans

[3.82,\,4.10]\times 10^{-7}

—all narrower than the between-phase performance differences by more than one order of magnitude. (b) Off-diagonal

R^{2}

versus AR bin median for Phase 4 (10 equi-populated bins,

n=400

each): gap is largest at near-isotropic AR (

\Delta R^{2}=0.181

) and narrows sharply to

<0.015

for

\mathrm{AR}>1.5

, with a modest secondary widening at high AR (

\Delta R^{2}=0.015

at AR

\approx 5.87

). (c) Component-wise bootstrap

R^{2}

distributions: narrow, symmetric densities for all four components; 95 % BCa CIs are

R^{2}_{K_{xx}}\in[0.9963,\,0.9971]

and

R^{2}_{K_{xy}}\in[0.9736,\,0.9779]

. (d) Cross-phase off-diagonal RRMSE versus AR bin: Phase 4 reduces error at every AR level, with the largest absolute improvement in the moderate-AR regime (

1.5<\mathrm{AR}<3

Appendix figures & tables13 assets

Supplementary material from the paper’s appendix.

Appendix

Property	Train	Validation	Test
Partitioning
Number of samples	16,000	4,000	4,000
Image dimensions	$128\times 128$	$128\times 128$	$128\times 128$
Microstructure geometry
Strictly binary (no. non-binary pixels)	True (0)	True (0)	True (0)
Porosity $\phi$ (mean $\pm$ std)	$0.711\pm 0.124$	$0.710\pm 0.125$	$0.713\pm 0.124$

Appendix

Table 3: Comprehensive dataset statistics across training, validation, and test splits. All permeability values in lattice units.

K_{\mathrm{diag}}

: diagonal elements (

K_{xx}

K_{yy}

);

K_{\mathrm{off}}

: off-diagonal elements (

K_{xy}

K_{yx}

);

\lambda_{\min}

\lambda_{\max}

: smallest and largest eigenvalues of

\mathbf{K}

;

\mathrm{AR}

: anisotropy ratio

\lambda_{\max}/\lambda_{\min}

;

\delta

: off-diagonal dominance

|K_{\mathrm{off}}|/(|K_{\mathrm{diag}}|+|K_{\mathrm{off}}|)

Split	Kozeny-Carman		Power law			$\rho_{s}$
	$C$	$R^{2}$	$C$	$n$	$R^{2}$
Train	0.0242	0.789	1.337	4.81	0.777	0.967
Val	0.0240	0.791	1.337	4.84	0.776	0.967
Test	0.0243	0.790	1.371	4.89	0.791	0.968

Appendix

Table 4: Porosity-permeability model fits on mean diagonal permeability

\bar{K}_{\mathrm{diag}}=\tfrac{1}{2}(K_{xx}+K_{yy})

across data splits. KC: Kozeny-Carman fit

\bar{K}=C\phi^{3}/(1-\phi)^{2}

; PL: power-law fit

\bar{K}=C\phi^{n}

\rho_{s}

: Spearman rank correlation between

\phi

and

\bar{K}_{\mathrm{diag}}

Pair	Variable	KS stat	$p$ -value	JSD
Train vs. Val	$K_{xx}$	0.0154	0.432	0.00353
	$K_{xy}$	0.0174	0.286	0.00305
	$K_{yy}$	0.0134	0.605	0.00368
	$\phi$	0.0141	0.547	0.00355
Train vs. Test	$K_{xx}$	0.0141	0.541	0.00312
	$K_{xy}$	0.0106	0.864	0.00385

Appendix

Table 5: Cross-split distributional consistency. KS: two-sample Kolmogorov-Smirnov statistic and

p

-value (reject if

p<0.05

). JSD: Jensen-Shannon divergence (bounded in

[0,\ln 2]

; values near 0 indicate near-identical distributions). All KS tests fail to reject the null hypothesis of equal distributions (

p>0.05

Split	Mahalanobis ( $>p_{99}$ )	High $\phi$ ( $>p_{99}$ )	Low $\phi$ ( $<p_{1}$ )	High AR ( $>p_{95}$ )
Train ( $n=16{,}000$ )	160 (1.00 %)	159 ( $\phi>0.896$ )	160 ( $\phi<0.396$ )	800 (AR $>4.59$ )
Val ( $n=4{,}000$ )	40 (1.00 %)	40 ( $\phi>0.896$ )	40 ( $\phi<0.390$ )	200 (AR $>4.63$ )
Test ( $n=4{,}000$ )	40 (1.00 %)	39 ( $\phi>0.896$ )	40 ( $\phi<0.401$ )	200 (AR $>4.47$ )

Appendix

Table 6: Outlier summary across data splits. Mahalanobis: samples beyond the 99th-percentile Mahalanobis distance in the joint tensor space. Extreme

\phi

: samples in the top or bottom 1 % of the marginal porosity distribution. High AR: samples in the top 5 % of the anisotropy ratio distribution.

Stage	Spatial resolution	Channels	MaxViT blocks	Attention regime
1	$32\times 32$	64	2	Block ( $4\times 4$ blocks) + Grid
2	$16\times 16$	128	6	Block ( $2\times 2$ blocks) + Grid
3	$8\times 8$	256	14	Full self-attention (spatial $\leq P$ )
4	$4\times 4$	512	3	Full self-attention (spatial $\leq P$ )

Appendix

Table 7: MaxViT-Base stage configuration for

128\times 128

input. Block size

P=8

is fixed across all stages.

Phase	Epochs	Init. from	LR max	LR min	WU	WD	BS	Pat.
2	600	ImageNet	$10^{-4}$	$10^{-7}$	50	0.05	32	100
3	600	Phase 2	$10^{-4}$	$10^{-5}$	50	0.01	32	100
4	600	Phase 3	$10^{-4}$	$10^{-6}$	50	0.01	32	150

Appendix

Table 8: Training configuration by phase. LR: learning rate; WD: weight decay; BS: batch size; Pat.: early stopping patience (epochs, monitoring validation

R^{2}

); WU: warmup duration (epochs).

Epoch range	Trainable components
0–50	Regression head only (backbone frozen)
50–150	Regression head + backbone Stages 3–4
	(Stages 1–2 and stem frozen)
150–600	Full model (all parameters)

Appendix

Table 9: Progressive unfreezing schedule in Phase 2.

Phase	$\lambda_{\mathrm{sym}}$	$\lambda_{\mathrm{pos}}$	$\lambda_{\mathrm{offdiag}}$	$(w_{\mathrm{d}},\,w_{\mathrm{o}})$
2	0.10	0.05	$-$	$(1.0,\,1.0)$
3–4	0.20	0.05	1.0	$(1.0,\,1.5)$

Appendix

Table 10: Physics-aware loss-function weights by training phase.

\lambda_{\mathrm{offdiag}}

is not active (

-

) in Phase 2.

Transformation	$\mathbf{P}$	$\mathbf{K}^{\prime}$
Identity	$\bigl[\begin{smallmatrix}1&0\\ 0&1\end{smallmatrix}\bigr]$	$\bigl[\begin{smallmatrix}a&b\\ b&c\end{smallmatrix}\bigr]$
Rotation $90^{\circ}$	$\bigl[\begin{smallmatrix}0&-1\\ 1&\phantom{-}0\end{smallmatrix}\bigr]$	$\bigl[\begin{smallmatrix}c&-b\\ -b&a\end{smallmatrix}\bigr]$
Rotation $180^{\circ}$	$\bigl[\begin{smallmatrix}-1&\phantom{-}0\\ 0&-1\end{smallmatrix}\bigr]$	$\bigl[\begin{smallmatrix}a&b\\ b&c\end{smallmatrix}\bigr]$
Rotation $270^{\circ}$	$\bigl[\begin{smallmatrix}0&1\\ -1&0\end{smallmatrix}\bigr]$	$\bigl[\begin{smallmatrix}c&-b\\ -b&a\end{smallmatrix}\bigr]$
Horizontal flip	$\bigl[\begin{smallmatrix}-1&0\\ 0&1\end{smallmatrix}\bigr]$	$\bigl[\begin{smallmatrix}a&-b\\ -b&c\end{smallmatrix}\bigr]$
Vertical flip	$\bigl[\begin{smallmatrix}1&\phantom{-}0\\ 0&-1\end{smallmatrix}\bigr]$	$\bigl[\begin{smallmatrix}a&-b\\ -b&c\end{smallmatrix}\bigr]$

Appendix

Table 11:

D_{4}

transformation matrices

\mathbf{P}

and the corresponding permeability tensor transformations

\mathbf{K}^{\prime}=\mathbf{P}\mathbf{K}\mathbf{P}^{T}

. Entries use

a=K_{xx}

b=K_{xy}=K_{yx}

c=K_{yy}

Comp.	Metric	Observed	Boot. mean	CI low	CI high
$K_{xx}$	$R^{2}$	0.99674	0.99673	0.99626	0.99708
	RMSE	$1.955\times 10^{-2}$	$1.956\times 10^{-2}$	$1.835\times 10^{-2}$	$2.120\times 10^{-2}$
	MAE	$1.272\times 10^{-2}$	$1.272\times 10^{-2}$	$1.224\times 10^{-2}$	$1.316\times 10^{-2}$
	MAPE (%)	7.09	7.08	6.64	7.61
	RRMSE (%)	5.25	5.25	4.96	5.67
$K_{xy}$	$R^{2}$	0.97583	0.97579	0.97359	0.97792

Appendix

Table 12: Component-wise bootstrap statistics for Phase 4 (

B=1{,}000

, 95 % BCa CIs). RRMSE normalized by component mean absolute value (Equation 23 ).

AR bin	$R^{2}_{\mathrm{diag}}$	$R^{2}_{\mathrm{off}}$	$\Delta R^{2}$	RRMSE ${}_{K_{xx}}$ (%)	RRMSE ${}_{K_{xy}}$ (%)	RRMSE ${}_{K_{yy}}$ (%)
1.115	0.9966	0.8152	0.1814	3.68	60.44	3.85
1.233	0.9968	0.9148	0.0820	3.98	39.46	3.90
1.334	0.9966	0.9690	0.0276	4.32	24.32	3.78
1.439	0.9972	0.9756	0.0216	4.03	21.92	3.91
1.556	0.9963	0.9830	0.0133	5.40	20.71	5.03
1.699	0.9963	0.9836	0.0127	6.59	23.10	5.17

Appendix

Table 13: Anisotropy-stratified prediction performance for Phase 4. Ten equi-populated bins (

n=400

each) spanning the 2nd–98th percentile of the AR distribution.

\Delta R^{2}=R^{2}_{\mathrm{diag}}-R^{2}_{\mathrm{off}}

Comp.	$R^{2}$	RMSE	$\bar{\sigma}$	$\tilde{\sigma}$	$\sigma_{p95}$	$\rho_{s}(\sigma,\|\varepsilon\|)$
$K_{xx}$	0.9967	$1.966\times 10^{-2}$	$1.263\times 10^{-2}$	$9.07\times 10^{-3}$	$3.34\times 10^{-2}$	0.229
$K_{xy}$	0.9756	$1.031\times 10^{-2}$	$5.15\times 10^{-3}$	$3.71\times 10^{-3}$	$1.33\times 10^{-2}$	0.304
$K_{yx}$	0.9756	$1.031\times 10^{-2}$	$5.15\times 10^{-3}$	$3.71\times 10^{-3}$	$1.33\times 10^{-2}$	0.304
$K_{yy}$	0.9967	$1.974\times 10^{-2}$	$1.266\times 10^{-2}$	$9.08\times 10^{-3}$	$3.29\times 10^{-2}$	0.235
Global	—	—	$8.90\times 10^{-3}$	$6.65\times 10^{-3}$	—	—

Appendix

Table 14: Per-component MC-Dropout uncertainty statistics for Phase 4 (

T=100

N=4{,}000

\bar{\sigma}

: mean predictive standard deviation;

\tilde{\sigma}

: median;

\sigma_{p95}

: 95th-percentile uncertainty;

\rho_{s}(\sigma,|\varepsilon|)

: Spearman rank correlation between predictive uncertainty and absolute error.

Metric		Phase 4	Phase 3	Phase 2
A. Global accuracy
	$R^{2}$ (component-averaged)	0.98628	0.98476	0.98429
	95 % BCa CI	[0.98516, 0.98738]	[0.98333, 0.98604]	[0.98304, 0.98563]
	Bootstrap $\sigma$	0.00059	0.00068	0.00070
	$R^{2}$ (variance-weighted)	0.99604	0.99452	0.99403
	MSE	$\mathbf{2.446\times 10^{-4}}$	$2.880\times 10^{-4}$	$3.226\times 10^{-4}$

Appendix

Table 15: Extended performance metrics across three consecutive training phases evaluated on the held-out test set (

N=4{,}000

). 95 % BCa bootstrap confidence intervals (

B=1{,}000

resamples) are shown in brackets; bootstrap standard deviations (

\sigma

) quantify estimator variability. Predictive uncertainty is estimated via Monte Carlo Dropout (

T=100

stochastic forward passes). Bold values mark the best result per row. Phase 2: supervised baseline with D4 augmentation; Phase 3: advanced augmentation and enhanced physics-aware loss; Phase 4: frozen backbone, FiLM porosity conditioning, SWA/EMA ensemble. Abbreviations — KGE: Kling–Gupta efficiency [ 32 ] ;

d

: Willmott index of agreement [ 78 ] ; MCE: mean calibration error (lower is better calibrated); UE-corr

\rho

: Spearman correlation between predicted standard deviation and absolute error;

\Delta R^{2}_{\mathrm{high\text{-}AR}}

: diagonal–off-diagonal

R^{2}

gap for the 500 most anisotropic test samples. MAPE for

K_{xy}

K_{yx}

reflects near-zero denominators in quasi-isotropic samples; RRMSE is the preferred relative metric in this regime (Appendix G ).

Materials Property Prediction

Materials property prediction uses data and physical models to estimate how composition, structure, and processing determine a material’s behavior.

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Topological Data Analysis combined with Machine Learning for Predicting Permeability of Porous Media

Ebru Dagdelen, Catherin Neena Lalu, Aakash Karlekar, Manav Arora+4

Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ · Department of Physics, New Jersey Institute of Technology, Newark, NJ

Materials Property Prediction Porous Media Flow

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Flow in porous media is difficult to address using standard analytical or numerical methods due to its complexity. However, since synthetic representations of porous media are easy to produce and data from physical experiments are becoming more widely available, the problem is well-suited to studies that include machine learning (ML) techniques. We discuss a number of features that can be extracted from such data, and their utility as input variables into a standard ML algorithm. These features include structural measures describing the geometry of the porous media, topological measures describing the connectivity, and network measures obtained by modeling the porous media as simplified pore networks. These features enable the prediction of the permeability of the considered (synthetic) porous materials using ML techniques that also leverage the separately computed exact permeability (ground truth). Comparing results obtained using different input variables helps develop a better understanding of the utility of various measures for predicting permeability based on the porous media structure. We show, in particular, that topological data analysis (TDA) provides a useful set of features that can be easily combined with ML to yield meaningful results.

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Materials Property Prediction

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Topological Data Analysis combined with Machine Learning for Predicting Permeability of Porous Media

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