Hard-constraint physics-residual networks enable robust extrapolation for hydrogen crossover prediction in PEM water electrolyzers

Hydrogen crossover in polymer electrolyte membrane water electrolysis poses a critical safety and efficiency bottleneck for scalable green hydrogen production. While machine learning offers real-time monitoring capabilities, conventional data-driven newral networks (Pure NNs) and soft-constraint physics-informed neural networks (Standard PINNs) suffer from inherent optimization conflicts and fail catastrophically when extrapolating beyond sparse training conditions. Here, we present a hard-constraint physics-residual network (PR-Net) that embeds analytical transport equations -- Henry's law, Fick's diffusion, and Faraday's law -- as a deterministic computational backbone, restricting the neural network to learn only systematic physical deviations. Across 184 experimental points spanning six membrane types and operating conditions of 25--85$^{\circ}$C, 1--200~bar, and 0.05--5.0 A cm$^{-2}$, this architecture intrinsically resolves gradient conflicts, yielding $R^{2} = 99.57 \pm 0.16\%$ with a 39-fold reduction in training variance compared to purely data-driven models ($R^{2} = 96.47 \pm 6.20\%$). Crucially, the PR-Net breaks the extrapolation barrier, maintaining $R^{2} > 97\%$ at extreme cathode pressures up to 200~bar -- a 2.5-fold extrapolation beyond the training domain where Standard PINN severely degrades ($R^{2} = 72.2\%$) and Pure NN collapses ($R^{2} = 58.7\%$). Furthermore, the learned residuals autonomously capture temperature-induced membrane swelling (Spearman's $\rho = 0.506$, $p < 0.001$) and identify the non-linear transport regime transition near 0.23 A cm$^{-2}$, without explicit programming. Delivering millisecond-level inference on edge hardware, the PR-Net establishes a highly reliable, generalizable foundation for adaptive safety control and predictive maintenance in high-pressure electrochemical energy systems.