cs.LGMar 30, 2026

Physics-Informed Neural Networks for Predicting Hydrogen Sorption in Geological Formations: Thermodynamically Constrained Deep Learning Integrating Classical Adsorption Theory

Authors: Mohammad Nooraiepour, Mohammad Masoudi, Zezhang Song, Helge Hellevang

Organizations: Faculty of Mathematics & Natural Sciences, University of Oslo, P.O. Box 1047 Blindern, 0316 Oslo, Norway · SINTEF Industry, Applied Geoscience Department, 7465 Trondheim, Norway · College of Geosciences, China University of Petroleum, Beijing 102249, China

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Abstract

Accurate prediction of hydrogen sorption in fine-grained geological materials is essential for evaluating underground hydrogen storage capacity, assessing caprock integrity, and characterizing hydrogen migration in subsurface energy systems. Classical isotherm models perform well at the individual-sample level but fail when generalized across heterogeneous populations, with the coefficient of determination collapsing from 0.80-0.90 for single-sample fits to 0.09-0.38 for aggregated multi-sample datasets. We present a multi-scale physics-informed neural network framework that addresses this limitation by embedding classical adsorption theory and thermodynamic constraints directly into the learning process. The framework utilizes 1,987 hydrogen sorption isotherm measurements across clays, shales, coals, supplemented by 224 characteristic uptake measurements. A seven-category physics-informed feature engineering scheme generates 62 thermodynamically meaningful descriptors from raw material characterization data. The loss function enforces saturation limits, a monotonic pressure response, and Van't Hoff temperature dependence via penalty weighting, while a three-phase curriculum-based training strategy ensures stable integration of competing physical constraints. An architecture-diverse ensemble of ten members provides calibrated uncertainty quantification, with post-hoc temperature scaling achieving target prediction interval coverage. The optimized PINN achieves R2 = 0.9544, RMSE = 0.0484 mmol/g, and MAE = 0.0231 mmol/g on the held-out test set, with 98.6% monotonicity satisfaction and zero non-physical negative predictions. Physics-informed regularization yields a 10-15% cross-lithology generalization advantage over a well-tuned random forest under leave-one-lithology-out validation, confirming that thermodynamic constraints transfer meaningfully across geological boundaries.

Figures & tables

ModelParam.Pressure RangeKey FeaturesAdvantagesBest ApplicationParameter Identifiability
Henry’s Law1Very lowLinear, dilute gasThermodynamic basis, simpleInitial slope analysisHigh
Langmuir2Low to mediumMonolayer, uniform sitesPhysical meaning, well-definedShales (monolayer)High
Freundlich2Wide rangeEmpirical power-lawHeterogeneity captureCoals (microporous)Medium
BET2Low relativeMultilayer formationSurface area analysisN 2 characterizationMedium
Temkin2MediumHeat decreases linearlyInteraction energeticsCharged clay mineralsMedium
Toth3Wide rangeModified saturationExtended pressure fitsHeterogeneous surfacesMedium
Table 1: Comparison of classical adsorption isotherm models for hydrogen sorption prediction. The table summarizes nine fundamental isotherm models, presenting their applicable pressure ranges, distinguishing features, primary advantages, optimal lithological applications, and parameter identifiability characteristics.
LithologyMean R 2Median R 2RMSETop ModelExcellent Fits
   (mmol/g)(Preference)(%)
Clays0.903 ±\pm 0.1530.9730.029 ±\pm 0.044Sips (57.6%)63.6
Shales0.836 ±\pm 0.2670.9680.030 ±\pm 0.048Sips (52.4%)52.4
Coals0.807 ±\pm 0.2380.9760.026 ±\pm 0.035Sips (73.3%)40.0
Overall0.861 ±\pm 0.2130.9700.029 ±\pm 0.044Sips (59.4%)55.1
Secondary model preferences: Freundlich (26.1%), Redlich-Peterson (7.2%), Temkin (4.3%), BET (1.4%), D-R (1.4%)
Table 2: Summary of classical isotherm model performance for individually fitted samples across geological lithologies. Statistics presented as mean ±\pm standard deviation. Model preference indicates the percentage of samples best represented by each theoretical framework after multi-criteria selection (AIC, BIC, physics score).
Figure 1: Classical isotherm model-fitting performance across individual geological samples of clays, shales, and coals. Nine classical models were fitted per isotherm. The best model was selected via the lowest AIC. High per-sample accuracy shows effective capture of local adsorption, yet fails to generalise across samples or aggregates (see Fig. 2 ). (a) Best-fit model R2R^{2} distributions by lithology (violin plots with interquartile boxes and data points). Dashed lines mark thresholds: R2=0.95R^{2}=0.95 (excellent), 0.850.85 (good), 0.700.70 (acceptable). (b) Model preference by lithology (% of samples with lowest AIC). Sips model dominates overall (59.4% of best fits). (c) Sample quality tiers by lithology: Excellent ( R2>0.95R^{2}>0.95 ), Good ( 0.850.85 – 0.950.95 ), Fair ( 0.700.70 – 0.850.85 ), Poor ( R2<0.70R^{2}<0.70 ). Percentages show tier shares within each lithology. (d) Sips model parameter distributions: maximum capacity qmax⁡q_{\max} (mmol g -1 ) and heterogeneity exponent nsn_{s} (reference line at ns=1n_{s}=1 , Langmuir limit). (e) Gibbs free energy of adsorption ( ΔG\Delta G , kJ mol -1 ) by lithology, derived from isosteric estimates. Shaded bands indicate physisorption strength: weak (0 to −10-10 ), moderate ( −10-10 to −20-20 ), strong ( <−20<-20 ).
Figure 2: Failure of classical isotherm models in aggregated, cross-sample generalisation. While individual-sample fits achieve median R2=0.97R^{2}=0.97 (Fig. 1 ), aggregating data within each lithology and re-fitting yields 80–90% mean reduction in R2R^{2} , motivating the physics-informed neural network approach. (a) Comparison of individual-sample R2R^{2} , aggregated R2R^{2} , and cross-validation R2R^{2} by lithology and overall. Percentage reductions annotate performance collapse. (b) Aggregated R2R^{2} heatmap for nine classical models plus fourth-order polynomial across four groupings (clays, shales, coals, all combined). Colour scale 0–0.45; best model per group highlighted (bold border). (c) Bootstrap parameter uncertainty (95% CI error bars) for dominant model per lithology: Redlich–Peterson (clays), Sips (shales), Langmuir (coals). Fold-width annotations (e.g. ×102\times 10^{2} ) indicate parameter instability. (d) Five-fold cross-validation R2R^{2} by lithology. Shale folds reach negative values, showing worse-than-mean performance. (e) Residual bias across six pressure bins ( << 10 to >> 200 bar). S-shaped pattern (under-prediction at low/high pressures, over-prediction at intermediate) reveals structural misspecification. Durbin–Watson statistics (1.18–1.37) confirm positive residual autocorrelation.
DatasetBest ModelTypeTraining R 2CV R 295% CI
    (mean ±\pm std)(Bootstrap)
ClaysPolynomial-4Math0.1720.165 ±\pm 0.013[0.151, 0.199]
 Redlich-PetersonClassical0.1610.154 ±\pm 0.017[0.139, 0.189]
 SipsClassical0.1460.139 ±\pm 0.017[0.126, 0.169]
ShalesPolynomial-4Math0.042−- 0.017 ±\pm 0.067[0.019, 0.085]
 HillClassical0.041−- 0.014 ±\pm 0.065[0.022, 0.076]
Table 3: Aggregated data generalization performance comparing classical isotherm models and mathematical equations across lithologies. Bootstrap confidence intervals computed from 1,000 iterations. Cross-validation statistics from five-fold splitting. All models demonstrate severe generalization failure, with R 2 values 70–95% lower than individual sample fits.
Figure 3: Property–uptake relationships for hydrogen sorption across 224 measurements in clays ( n=123n=123 ), shales ( n=39n=39 ), and coals ( n=62n=62 ). Pearson correlation analysis revealed 23 significant relationships ( ∣r∣>0.3|r|>0.3 , p<0.05p<0.05 ; 2 clay, 12 shale, 9 coal), with bootstrap resampling (1000 iterations) for uncertainty quantification. Results highlight lithology-specific controls — surface area in clays, organic content in shales, and rank-dependent aromaticity in coals — motivating the lithology-branched PINN architecture (Section 3.5 ). (a) Hydrogen uptake distributions by lithology (violin plots with interquartile boxes, medians, and data points). Coals showed highest mean uptake ( 0.378±0.2050.378\pm 0.205 mmol g -1 , CV 54%), shales the widest range ( 0.282±0.4500.282\pm 0.450 mmol g -1 ), and clays the greatest relative variability ( 0.232±0.2760.232\pm 0.276 mmol g -1 , CV 119%). (b) Significant property–uptake correlations ( ∣r∣>0.3|r|>0.3 , p<0.05p<0.05 ), grouped by lithology and sorted by decreasing ∣r∣|r| . Bar color indicates sign (green = positive, red = negative); values show Pearson rr and valid nn . (c) Bootstrap 95% confidence intervals for the three strongest correlations per lithology (1000 iterations). Wide CI for coal micropore volume ( r=0.650r=0.650 , 95% CI [0.118, 0.894], n=13n=13 ) reflects data sparsity limiting model performance. (d) Dominant driver of uptake for each lithology with linear regression and 95% bootstrap confidence bands (500 iterations): specific surface area (clays, r=0.598r=0.598 , n=121n=121 ), total organic carbon (shales, r=0.755r=0.755 , n=39n=39 ), fixed carbon (coals, r=0.711r=0.711 , n=48n=48 ). (e) Number of positive ( r>0.3r>0.3 ) and negative ( r<−0.3r<-0.3 ) significant correlations by lithology. Shales show the most diverse pattern (12 total: 7 positive, 5 negative), clays only positive (2 total), and coals predominantly positive (9 total: 6 positive, 3 negative).
Figure 4: Supervised predictive modelling and Random Forest feature importance for hydrogen uptake across three lithologies, providing performance baselines and feature priorities for PINN input layer design. Three model classes were tested — linear regression, Ridge regression ( α=1.0\alpha=1.0 ), and Random Forest (100 estimators, max depth 5), using 5-fold cross-validation. (a) Five-fold cross-validation R2R^{2} (mean ±\pm SD) by model and lithology (grouped bars with per-fold error bars). Red zone indicates worse-than-mean performance. Ridge markedly improves shale R2R^{2} from −1.130-1.130 (linear) to 0.5510.551 due to multicollinearity. Random Forest yields the highest performance overall (clay CV R2=0.523R^{2}=0.523 ; coal CV R2=0.352R^{2}=0.352 ). (b) Random Forest feature importance (mean decrease in impurity) by lithology (stacked horizontal bars). Values ≥5%\geq 5\% labelled. Clays are dominated by specific surface area (80.9%). Shales show distributed importance led by TOC (52.7%), langite (15.4%), average pore diameter (13.5%), and berlinite (11.2%). Coals are led by vitrinite reflectance (%R o , 23.2%) and volatile matter (22.0%). (c) Per-fold R2R^{2} variability (box-and-strip plots) for each model–lithology combination. Red zone marks negative performance. Shale linear regression shows extreme instability (some folds R2<−1.8R^{2}<-1.8 ), while Random Forest remains consistently positive. (d) Property data completeness (%) vs. absolute Pearson ∣r∣|r| for significant pairs ( p<0.05p<0.05 , ∣r∣>0.2|r|>0.2 ), with point size ∝n\propto\sqrt{n} and colour by lithology. Purple zone ( << 30% completeness) highlights coal micropore volume (21% complete, ∣r∣=0.650|r|=0.650 ), the strongest predictor yet least measured — a key challenge for coal PINN development. (e) PINN input feature priorities by lithology (lollipop chart), coloured by mechanism: blue (structural: surface area, pores), green (chemical: organics, mineral proxies), orange (rank: coalification indicators). Clay branches prioritize surface area; shale branches require TOC and pore metrics; coal branches emphasize rank indicators (%R o , volatile matter) plus texture.
Figure 5: Ensemble feature selection, bootstrap stability, cross-validation performance, and stratified dataset preparation. Four methods (Pearson correlation, mutual information regression, Random Forest importance, FF -statistic) were applied to the scaled feature matrix; final selection used majority voting. (a) Top 20 features ranked by mean bootstrap importance (absolute Pearson rr , 1000 iterations), shown as horizontal bars with 95% CI error bars. Colours indicate category (thermodynamic, pore structure, surface chemistry, interaction, kinetic, molecular sieving, classical model-derived), highlighting multi-domain predictive power. (b) Feature importance vs. bootstrap stability scatter: mean bootstrap importance ( xx -axis) vs. 95% CI width ( yy -axis, instability measure); point size reflects ensemble vote count (1–4). Green quadrant marks high-importance, low-instability features preferred for PINN input. Dashed lines show median thresholds. (c) Five-fold cross-validation performance of Random Forest baseline on selected features: R2R^{2} (blue, left axis) and RMSE (mmol g -1 , orange, right axis) per fold. Dashed lines indicate cross-fold means. (d) Lithology-stratified train (70%), validation (15%), and test (15%) splits via StratifiedShuffleSplit. Stacked bars show sample counts per lithology; percentages confirm proportional representation across partitions. (e) Kernel density estimates of H 2 uptake for each split and overall dataset. Two-sample Kolmogorov–Smirnov test confirms statistically consistent target distributions between training and test splits.
LithologyR 2RMSEMAEMax ErrorMBE
  (mmol/g)(mmol/g)(mmol/g)(mmol/g)
Clays0.96570.04880.02490.3406−- 0.0022
Shales0.82820.03250.01700.2244−- 0.0163
Coals0.76660.07770.02970.3816−- 0.0218
Table 4: Lithology-specific PINN performance on the held-out test set.
Figure 6: Prediction performance of the optimised multi-scale PINN on the held-out test set ( n=286n=286 ; clays n=185n=185 , shales n=36n=36 , coals n=65n=65 ), shown via parity plots of predicted vs. measured H 2 uptake. Overall metrics: R2=0.9544R^{2}=0.9544 (95% bootstrap CI [0.948, 0.961], 10,000 resamples), RMSE = 0.0484 mmol g -1 , MAE = 0.0231 mmol g -1 , MBE = −0.0081-0.0081 mmol g -1 (0.5% of mean uptake). Dashed diagonal indicates perfect agreement; all panels use shared axis scaling. (a) Overall parity plot for all 286 test points, coloured by lithology. Annotation reports global R2R^{2} , 95% CI, RMSE, MAE, and MBE. (b) Parity plot for clays ( R2=0.9657R^{2}=0.9657 , RMSE = 0.0488 mmol g -1 ). Highest accuracy reflects the largest training representation, surface-area-dominated mechanisms, and near-normal residuals (Shapiro-Wilk p=0.08p=0.08 ). (c) Parity plot for shales ( R2=0.8282R^{2}=0.8282 , RMSE = 0.0325 mmol g -1 ). Larger errors in high-TOC samples ( >3>3 wt%) and heteroscedasticity ( ρ=0.48\rho=0.48 ) indicate unresolved multi-scale heterogeneity. (d) Parity plot for coals ( R2=0.7666R^{2}=0.7666 , RMSE = 0.0777 mmol g -1 ). Ensemble stabilisation and physics constraints yield 92.6% calibration coverage despite the smallest training set and the widest rank range.
Figure 7: Residual diagnostics of the multi-scale PINN on the held-out test set, characterising error structure, distributions, and lithology-specific heteroscedasticity. Global residuals show mean −0.0081-0.0081 mmol g -1 , SD 0.05180.0518 mmol g -1 ; 95% of predictions lie within ±0.094\pm 0.094 mmol g -1 . (a) Residuals vs. predicted uptake, coloured by lithology. Global heteroscedasticity ( ρ=0.32\rho=0.32 , Breusch-Pagan p<0.001p<0.001 ); strongest in shales ( ρ=0.48\rho=0.48 ), weakest in clays ( ρ=0.18\rho=0.18 ) and coals ( ρ=0.12\rho=0.12 ). Shaded band marks ±\pm 95th percentile error. (b) Absolute error distributions by lithology. Reference lines: global median (0.0089), 90th (0.0534), and 95th (0.0935) mmol g -1 . Broader coal spread reflects rank variability and data limitation. (c) Residual frequency histogram with fitted Gaussian overlay. Modest skewness ( −- 0.22) and excess kurtosis (1.24) driven by rare high-uptake extremes; maximum absolute error 0.3816 mmol g -1 (single outlier). (d) Normal Q-Q plot of residuals. Upper-tail deviation matches observed skew; bulk residuals approximate normality, with Shapiro-Wilk rejection ( p<0.001p<0.001 ) attributable to tail behaviour.
Figure 8: Validation of physics-informed architecture via constraint satisfaction, ablation, benchmarking, and cross-validation stability, demonstrating performance gains and mechanistic contributions of each component. (a) Physics constraint violation rates: multi-scale PINN vs. unconstrained baseline. PINN achieves 0.0% non-negativity violations (vs. 8.4%), 1.4% upper-bound violations (vs. 12.6%), 98.6% monotonicity compliance (vs. 84.2%), and 87.3% saturation consistency. (b) Ablation of architectural components on test R2R^{2} : MLP baseline (0.921), +dropout ensemble (0.938), +architecture-diverse ensembling (0.947), +physics constraints (full PINN 0.9544). (c) Benchmark comparison vs. baselines with paired tt -tests: PINN outperforms Random Forest ( p<0.001p<0.001 , d=0.25d=0.25 ), standard MLP ( p<0.001p<0.001 , d=0.53d=0.53 ), and classical isotherm models ( p<0.001p<0.001 , d=1.46d=1.46 ). (d) Five-fold stratified cross-validation R2R^{2} (fold values: 0.951, 0.947, 0.958, 0.944, 0.953; mean 0.9506±0.00530.9506\pm 0.0053 , CV 0.56%). Dashed line and band show mean ±\pm SD; permutation tests confirm key feature importance ( p<0.001p<0.001 ).
Figure 9: Generalisation and ensemble analysis of the multi-scale PINN: leave-one-lithology-out transfer, dropout sensitivity, multi-seed stability, and architecture-diverse ensemble benefit. (a) Leave-one-lithology-out R2R^{2} : coals held out (trained on clays+shales) R2=0.723R^{2}=0.723 (5.8% degradation); shales held out (trained on clays+coals) R2=0.794R^{2}=0.794 (4.1% degradation). PINN outperforms Random Forest by 10–15% relative generalisation. In-distribution R2R^{2} shown hatched. (b) R2R^{2} sensitivity to dropout rate pp (15 seeds per value). Optimal p=0.10p=0.10 (lower than typical); p=0.12p=0.12 degrades coal R2R^{2} by 22%, showing minority lithology vulnerability to over-regularisation. (c) Test R2R^{2} across five independent runs (different seeds). Overall mean 0.9467±0.00330.9467\pm 0.0033 (CV 0.35%); highest variance in coals, stabilised by ensemble aggregation to final R2=0.7666R^{2}=0.7666 . (d) R2R^{2} of ten architecture-diverse ensemble members (range 0.9369–0.9544) vs. aggregate. Diverse ensembling reduces inter-member correlation ( r=0.9980r=0.9980 ) and improves performance by +0.88% over member average.
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