Physics-guided AI tracks fuel cell ageing with watt-level precision

A hydrogen fuel cell is a deceptively simple object: hydrogen in, electricity and water out. But the power that actually emerges from its stack depends on a tangle of coupled electrochemical and thermal processes that shift as the device ages, making real-time prediction notoriously difficult. A new open-access study published in Discover Electronics on 5 October 2026 reports a physics-informed deep learning framework that predicts the measured power output of a proton exchange membrane (PEM) fuel cell stack with a coefficient of determination of 0.969, a root-mean-square error of 0.295 watts, and a mean absolute percentage error of just 0.097 percent. The work, led by Muhammad Own Raza and colleagues including Nadeem Shaukat, was trained and evaluated against the IEEE PHM 2014 FC1 Ageing dataset, a publicly available log of 65,535 sequential measurements from a single laboratory fuel cell stack.

The central innovation lies in how physics enters the model. Rather than embedding electrochemical equations as penalty terms in the loss function, as many physics-informed neural network approaches do, the researchers computed first-principles quantities—the Nernst open-circuit voltage, activation overpotential from the Tafel approximation of the Butler-Volmer equation, ohmic losses from an effective membrane resistance, and concentration losses from a limiting-current formulation—and fed them to the network as engineered input features. The network itself is a Time-Series Mixer (TS-Mixer), a parallel dense architecture with layers of 256, 128, and 64 neurons, batch normalization, and dropout, trained with the Adam optimizer, a mean absolute error loss, early stopping, and adaptive learning-rate scheduling. The prediction target was the dataset’s directly measured stack power, calculated as total stack voltage multiplied by current, not a value derived from the same equations used to build the inputs.

That distinction matters more than it might first appear. In an earlier version of the framework, the authors trained the network to reproduce a stack power value computed from their own electrochemical equations applied to the dataset’s temperature, pressure, and current readings. Prompted by reviewer scrutiny, they recognized this amounted to training a network to approximate its own analytic generator function, which would overstate predictive capability. The revised design trains against the independent experimental observable while the physics-derived quantities serve only as structured inputs encoding domain knowledge. The authors are explicit that this feature-level approach is simpler to implement and audit but does not enforce physical consistency as a hard constraint, meaning the model could in principle produce predictions inconsistent with electrochemistry outside the training distribution.

A careful diagnostic of the dataset shaped the final design in another important way. The FC1 ageing test is a long-duration, single-operating-point experiment lasting roughly 497 hours, during which current density, temperature, and pressure barely move—the current density has a standard deviation of only 3.2 times ten to the minus four amperes per square centimeter around a mean of 0.702. What does change is the stack itself: measured voltage and power decline steadily with elapsed time, with a correlation of minus 0.955 between measured power and time, consistent with membrane and catalyst-layer degradation. Because the fixed-parameter electrochemical model contains no degradation term, its predicted cell voltage correlates with the measured stack voltage at only 0.235. The researchers therefore added cumulative operating time, and its product with current density, as two additional input features—a physically motivated choice, since an onboard management system always has access to operating hours and degradation rate is known to depend on operating current.

The ablation study quantifies exactly how much each ingredient contributes, and the results are refreshingly honest about what is doing the work. Compared with the naive baseline of predicting from elapsed time alone, the full physics-informed feature set improves test R-squared by roughly 3.8 percentage points and reduces RMSE from 0.437 watts to 0.295 watts.

To rule out that the reported accuracy reflects data leakage or an artifact of a favorable train-test split, the team benchmarked the TS-Mixer against six alternatives—LSTM, GRU, temporal convolutional network, Transformer, support vector regression, and ordinary linear regression—on identical data, features, and an 80/20 split. The TS-Mixer achieved the best R-squared among the tested architectures at 0.963 in that comparison, followed by support vector regression at 0.939 and the Transformer at 0.909, while the recurrent architectures performed markedly worse, plausibly because the target signal is dominated by a slow, near-monotonic degradation trend rather than the short-range sequential dependencies recurrent units are designed to exploit. Notably, linear regression was a fairly strong baseline at 0.919, suggesting a meaningful share of the predictive signal is close to linear once time-degradation and physics-derived features are supplied. Five-fold cross-validation over the complete dataset gave a mean test R-squared of 0.967 plus or minus 0.001, mean RMSE of 0.302 plus or minus 0.006 watts, and mean MAE of 0.228 plus or minus 0.002 watts, closely matching the single split and indicating no meaningful overfitting, with a near-zero train-test gap in R-squared of minus 0.0004.

Interpretability analyses on the retrained model reinforce the same picture. Permutation importance, computed as the drop in test R-squared when each feature is independently shuffled, shows the time-current-density interaction term dominating with a drop of 0.637, followed by elapsed time at 0.580. Among the remaining features, temperature contributes the most at 0.081, then the Nernst open-circuit voltage at 0.052 and activation overpotential at 0.051, while pressure contributes the least at 0.006. SHAP analysis with KernelExplainer corroborates this ranking, giving the time-current interaction and elapsed time the largest mean absolute SHAP values at 0.105 and 0.098 respectively, followed by temperature, activation overpotential, and the Nernst voltage, with pressure lowest. LIME explanations for three representative test instances similarly identify the time-related features as the dominant local contributors, with the electrochemical loss terms and raw operating variables providing secondary, instance-specific adjustments—a coherent, physically interpretable account of the model’s predictions at the level of individual samples.

Separately from the predictive validation, the underlying electrochemical equations were exercised as a sizing tool for an illustrative automotive-scale stack of 370 cells, each with an active area of 310 square centimeters. That simulation produced a stack power output averaging approximately 60,000 watts, or 60 kilowatts, with hydrogen consumption of roughly 3.02 kilograms per hour, thermodynamic properties matching theoretical values—an enthalpy change averaging minus 285,800 joules per mole and Gibbs free energy near 236,960 joules per mole—and stable operating parameters including a current density of approximately 0.701 amperes per square centimeter. The authors stress that this scaled simulation is deliberately kept separate from the rig-scale validation: it demonstrates that the equations produce physically reasonable stack-level behavior at automotive scale, but makes no claim about the physical configuration of the IEEE PHM 2014 test rig, which is a much smaller single-stack laboratory unit.

Deployment feasibility was assessed with unusual candor. The final TS-Mixer model contains 135,685 total parameters, of which 44,929 are trainable, corresponding to an on-disk weights file of just 577 kilobytes—comfortably within the memory limits of nearly all embedded and edge platforms. But single-sample inference latency, measured on a CPU-based Colab runtime rather than dedicated embedded hardware, averaged 77.6 milliseconds per sample when queried one at a time, which falls short of hard real-time budgets on constrained automotive-grade microcontrollers. Batching amortizes the fixed per-call framework overhead, dropping latency to 4.20 milliseconds per sample at batch size 16, 1.03 at batch size 64, and 0.12 at batch size 1024. The authors report both figures rather than a single optimistic number, noting that production deployment would likely require model quantization or pruning, batched or asynchronous inference scheduling, or GPU-accelerated edge compute such as Jetson-class hardware.

The limitations are stated as plainly as the results. All predictive findings derive from one fuel cell stack, and the cross-validation assesses robustness within that dataset only; generalization to an independent stack, different hardware, or a different operating regime is unproven. The dataset is an ageing test at an essentially fixed operating point, not a dynamic drive-cycle test, so no claim is made about performance under rapid transient loads such as a real driving cycle—an earlier manuscript version referencing synthetic transient load augmentation has been corrected. The electrochemical parameters are literature-typical constants, not fitted to this specific stack, and include no degradation term, which is why the physics features alone have limited predictive power here. With those caveats weighed, the study establishes a rigorously grounded pathway toward physics-informed monitoring of gradual fuel cell degradation using variables already available to an onboard management system, and the authors identify dynamic drive-cycle validation, explicit degradation modeling, PINN-style loss constraints, independent dataset validation, and multi-objective optimization as the next steps toward deployable supervisory control and digital-twin integration in fuel cell electric vehicles.

Subject of Research: Physics-informed deep learning for predicting PEM fuel cell stack power against measured data

Article Title: Physics informed deep learning for real time optimization of PEM fuel cells in electric vehicles

Article References: Raza, M. O., Hanzla, M., Shinwari, A. R., & Shaukat, N. (2026). Physics informed deep learning for real time optimization of PEM fuel cells in electric vehicles. Discover Electronics, 3(1), Article 143. https://doi.org/10.1007/s44291-026-00289-6

Image Credits: AI Generated

DOI: 10.1007/s44291-026-00289-6

Keywords: PEM fuel cells, physics-informed machine learning, TS-Mixer, IEEE PHM 2014 dataset, degradation prediction, hydrogen, electric vehicles, deep learning, SHAP, embedded inference, fuel cell electric vehicles, electrochemical modeling