Physics-Informed Neural Networks for Real-Time Gas Crossover Prediction in PEM Electrolyzers: First Application with Multi-Membrane Validation

TL;DR

This work presents the first application of physics-informed neural networks to predict hydrogen crossover in PEM electrolyzers, integrating mass conservation, diffusion, and thermodynamics within a compact network. By augmenting sparse experimental data with physics-constrained cubic splines and enforcing multiple transport constraints, the PINN achieves near-perfect predictive accuracy across six membranes and industrial operating ranges, while delivering sub-millisecond inference on desktop and edge hardware. The study demonstrates robust extrapolation through a physics-assisted fusion approach and provides well-calibrated uncertainty estimates via deep ensembles, enabling reliable real-time safety monitoring in gigawatt-scale deployments. The results establish a practical, hardware-agnostic framework that bridges physical rigor and computational efficiency, with clear implications for safer, more efficient green hydrogen production and potential extension to other electrochemical systems.

Abstract

Green hydrogen production via polymer electrolyte membrane (PEM) water electrolysis is pivotal for energy transition, yet hydrogen crossover through membranes threatens safety and economic viability-approaching explosive limits (4 mol% H$_2$ in O$_2$) while reducing Faradaic efficiency by 2.5%. Current physics-based models require extensive calibration and computational resources that preclude real-time implementation, while purely data-driven approaches fail to extrapolate beyond training conditions-critical for dynamic electrolyzer operation. Here we present the first application of physics-informed neural networks (PINNs) for hydrogen crossover prediction, integrating mass conservation, Fick's diffusion law, and Henry's solubility law within a compact architecture (17,793 parameters). Validated across six membranes under industrially relevant conditions (0.05-5.0 A/cm$^2$, 1-200 bar, 25-85°C), our PINN achieves exceptional accuracy (R$^{2}$ = 99.84% $\pm$ 0.15\%, RMSE = 0.0932% $\pm$ 0.0438%) based on five-fold cross-validation, with sub-millisecond inference times suitable for real-time control. Remarkably, the model maintains R$^2$ > 86% when predicting crossover at pressures 2.5x beyond training range-substantially outperforming pure neural networks (R$^2$ = 43.4%). The hardware-agnostic deployment, from desktop CPUs to edge devices (Raspberry Pi 4), enables distributed safety monitoring essential for gigawatt-scale installations. By bridging physical rigor and computational efficiency, this work establishes a new paradigm for real-time electrolyzer monitoring, accelerating deployment of safe, efficient green hydrogen infrastructure crucial for net-zero emissions targets.

Figures (11)

• Figure 1: Physics-informed neural network for hydrogen crossover prediction in PEM electrolyzers. (a) Parity plot demonstrating best model performance ($R^2 = 99.98\%$, RMSE = 0.0335%, MAPE = 1.50%) for 1,114 data points across six membrane types under operating conditions of 0.05--5 A cm$^{-2}$, 1--200 bar, and 25--85 $^\circ$C. The dashed line indicates perfect prediction, and gray bands represent $\pm$10% error margins. (b) Performance metrics of the best model: $R^2 = 99.98\%$, RMSE = 0.0335%, and MAPE = 1.50%. (c) Inference time benchmarking on three platforms: desktop CPU (0.18 ms), edge GPU (1.68 ms), and Raspberry Pi 4 (4.45 ms). The red dashed line marks the 2 ms real-time control threshold. (d) Residual distribution for the best model (mean = 0.0027%, $\sigma = 0.0334\%$), with 98.9% of predictions within $\pm$0.1% absolute error. The red curve represents the Gaussian kernel density estimate.
• Figure 2: Physics consistency validation demonstrating adherence to fundamental transport laws. (a) Model consistency showing excellent agreement between PINN predictions (blue circles, R$^2$ = 0.9998), physics-based calculations (orange squares, R$^2$ = 0.9947), and experimental measurements across 184 operating conditions spanning six membrane types, temperatures of 25--85°C, pressures of 1--200 bar, and current densities of 0.05--5.0 A/cm$^2$. Dashed line indicates perfect prediction; shaded regions represent ±5% (dark gray) and ±10% (light gray) error margins, with $>$95% of data points falling within ±10% error. (b) Pressure scaling demonstrates linear relationship consistent with Henry's law for dissolved hydrogen in Nafion 117 membrane at 25°C. Different colors represent current densities ranging from 0.07 to 1.0 A/cm$^2$ (n = 48 data points). Dashed lines show physics model predictions. (c) Current density effects revealing transition from diffusion-dominated (i $<$ 0.3 A/cm$^2$) to Faradaic-dominated (i $>$ 1.5 A/cm$^2$) transport regimes. Logarithmic scale highlights power-law behavior across three representative conditions: Nafion 117 (25°C, 80 bar, blue), Nafion 212 (60°C, 10 bar, orange), and Nafion 117-178µm (80°C, 20 bar, green). Shaded regions indicate dominant transport mechanisms.
• Figure 3: Extrapolation capabilities and uncertainty quantification of physics-informed neural networks for hydrogen crossover prediction. (a,b) Extrapolation performance of PINN and pure neural network models beyond the training pressure range. Models were trained on experimental data at 1, 6, 20, 40, and 80 bar (training range) and tested at 120, 160, and 200 bar (extrapolation range) for Nafion 117 membrane at 25°C. The fusion approach (0.5×ŷPINN + 0.5×ŷphysics) was applied only during extrapolation inference. Solid lines represent model predictions across continuous current density range (0.2--2.0 A cm$^{-2}$), with shaded regions indicating 95% confidence intervals derived from ensemble predictions (n = 100 models). Scatter points show experimental measurements. The PINN model (a) maintains high prediction accuracy even at 200 bar (R$^2$ = 0.864, RMSE = 1.49%), while the pure NN model (b) shows substantial degradation (R$^2$ = 0.434, RMSE = 3.04%). (c) Quantitative comparison of model performance metrics across extrapolation pressures. Left y-axis shows coefficient of determination (R$^2$, bars) and right y-axis shows root mean square error (lines with markers). Green annotations indicate percentage improvement of PINN over NN model. The performance gap widens progressively with extrapolation distance, reaching 99.0% improvement in R$^2$ at 200 bar. (d) Error distribution analysis using violin plots showing prediction residuals (experimental -- predicted) for both models. Horizontal dashed line indicates zero error. Box annotations display mean absolute error (MAE) for each pressure condition. The PINN model exhibits tighter error distributions with lower bias across all pressures, demonstrating superior generalization capability.
• Figure 4: Dataset Analysis #1: Comprehensive dataset analysis of hydrogen crossover in PEM water electrolysis under varying operating conditions.a, Relationship between H$_2$ crossover concentration and current density across eight independent studies (n = 184 data points). b, Membrane-specific H$_2$ crossover behavior for six membrane types under varying current densities (0.05--5.00 A cm$^{-2}$). c, Temperature dependence of mean H$_2$ concentration across the operating range of 25--85°C. Error bars represent standard deviation. d, Distribution of H$_2$ concentration as a function of operating pressure (1--200 bar). Box plots show median, interquartile range, and outliers. e, Comparative analysis of mean H$_2$ crossover concentration across studies, highlighting the predominant contribution from Wu 2025. Data compiled from Grigoriev 2011, Schalenbach 2013, Stahler 2020, Trinke 2017, Martin 2021, Garbe 2019, Wu 2025, and Bernt 2020.
• Figure 5: Dataset Analysis #2: Statistical dataset analysis and parameter correlations for hydrogen crossover in PEM electrolysis.a, Comparative analysis of mean H$_2$ crossover trends for six membrane types (Nafion 117, Nafion 212, Fuma Tech, Nafion D2021, Nafion 117 178$\mu$m, Nafion 212 51$\mu$m) as a function of current density (0--5 A cm$^{-2}$). b, Temperature-dependent H$_2$ concentration profiles at four operating temperatures (25°C, 60°C, 80°C, 85°C) across the full current density range. c, Violin plots illustrating the distribution and probability density of H$_2$ concentration within discrete current density bins, revealing data spread and median values at different operating conditions. d, Correlation matrix depicting relationships between key operational parameters (temperature, pressure, current density) and H$_2$ concentration. Pressure demonstrates the strongest positive correlation with H$_2$ concentration (r = 0.77).