Physics-informed distribution of relaxation times estimation and latent-space condition monitoring of solid oxide fuel and electrolysis cells from electrochemical impedance spectroscopy

The paper proposes a physics-informed convolutional neural network (CNN) framework for estimating the distribution of relaxation times (DRT) from electrochemical impedance spectroscopy (EIS) data, and for using the learned latent representation for condition monitoring of solid oxide fuel and electrolysis cells.

The core problem addressed is that estimating DRT from EIS is an ill-posed inverse problem, highly sensitive to regularisation choices. The authors state: Estimating the distribution of relaxation times (DRT) from electrochemical impedance spectroscopy (EIS) is an ill-posed inverse problem that is highly sensitive to regularisation choices. Traditional methods require hand-tuned regularisation, and per-spectrum neural network approaches vary implicit regularisation from measurement to measurement, making it difficult to attribute DRT differences to genuine electrochemical changes.

The proposed framework embeds the discretised relation between impedance and DRT directly into the training process: A discretised relation between impedance and the DRT is embedded in the training process, constraining the network to produce impedance-consistent distributions. The CNN takes the real and imaginary components of impedance as two input channels and outputs the DRT. The predicted DRT is then inserted into a discretised form of the governing integral equation, and the reconstructed impedance is compared with the measured spectrum to compute the training loss. The loss is defined as: L = MSE(Ẑ′, Z′) + MSE(Ẑ′′, Z′′) + λL L2, where λL = 106 penalises large inductive contributions. An auxiliary branch predicts series resistance Rs, and a trainable parameter represents inductive contribution L.

Key architectural choices include replacing transposed convolutions with upsampling followed by standard convolution to eliminate checkerboard artifacts: each transposed convolution was replaced by an upsampling layer followed by a standard convolution, producing smooth reconstructions without artificial structures. A grid search across 216 architectures identified a receptive field of 57 and bottleneck size of 160, yielding approximately 12,000 trainable parameters. The authors note: "the selected architecture was not optimized for a specific experiment. Instead, the same network configuration was subsequently applied to all investigated datasets without any dataset-specific redesign or hyperparameter tuning."

The framework was validated on three independent experimental datasets: (1) a 3600-hour SOFC stack monitoring campaign with 603 EIS spectra, (2) an operating condition study on an industrial-scale SOEC with 150 spectra under three conditions, and (3) a multi-regime degradation campaign with 1248 spectra over 2650 hours covering six phases. Reconstruction accuracy was quantified: the range-normalised RMSE remains below approximately 1.1%, with specific values of 0.291%, 1.062%, and 0.451% for the three datasets respectively.

Validation on synthetic two-ZARC spectra with analytically known DRTs showed the framework resolves overlapping relaxation processes. For two-decade separation, the network accurately reproduced the two-peak structure, though the higher-τ peak was shifted by 0.241 decades. For the more challenging one-decade separation, where the Nyquist spectrum appears as a single broad depressed arc, the network reproduced the correct two-peak structure with peak-position errors of only 0.065 and 0.025 decades.

Decoder probe analysis revealed that the learned latent representation is physically organised: the pooled position determines a coarse relaxation-time region, while the channel identity provides additional resolution within that region. Pooled positions 0 and 1 activate long-timescale processes (median peaks at approximately τ = 24 s and τ = 83 s), while positions 2 and 3 shift to shorter timescales (median peaks near τ = 3.5 s and τ = 0.3 s). This organisation emerged without explicit supervision.

For condition monitoring, Euclidean distances in latent space, defined as Dref(t) = ∥zt − zref∥2, reliably identified operational events. In the SOFC stack monitoring dataset, pronounced peaks coincide with known operational events during the experimental campaign, including fuel starvation events, H2 shortage shutdowns, power outages, and facility relocation. The authors state: "Increases in D10(t), seen as peaks in Figure 9, coincide with fuel starvation events, shutdowns, and other major perturbations, demonstrating that deviations from the nominal operating state are reflected directly and immediately in the latent representation."

For the operating condition study, element-wise absolute differences zt − z0 captured the same transitions identified by frequency-resolved KL divergence analysis in the original study: diffusion onset at 20 h, air-electrode changes at 40 h for Condition 1; two distinct transitions for Condition 2; and sustained recurring latent activity for Condition 3's pulsating instability.

For the multi-regime degradation campaign, the framework reproduced the phase-by-phase degradation trends using the same metric as the original study, ∆gt(log τ) = gt(log τ) − g1(log τ). The results showed: low-frequency DRT growth during co-electrolysis (P2), partial retreat upon returning to steam electrolysis (P4), periodic high-frequency modulation during reversible EC/FC operation (P5), and slow drift in the final phase (P6). The latent distance D10(t) independently traced the same trajectory: the slope steepens at the onset of co-electrolysis, relaxes upon recovery, oscillates through the EC/FC phase, and settles into a slow drift in P6.

The authors conclude: "Physics-informed learning can therefore replace the manual preprocessing pipeline that has long been the bottleneck in large-scale EIS diagnostics. A single, unified, and fully reproducible model is sufficient to extract physically meaningful DRTs, monitor system health, and generalise across the diverse operating conditions encountered in electrochemical energy conversion research."

Improvements to AI Systems:

1. Physics-constrained loss functions for inverse problems: Embed the governing discretized integral equation directly into the training loss (as done with the impedance–DRT relation) to enforce physical consistency, eliminating the need for hand-tuned regularization and reducing sensitivity to ill-posedness.

2. Architecture-level artifact suppression: Replace transposed convolutions with upsampling + standard convolution to eliminate checkerboard artifacts, ensuring smooth, physically plausible outputs without artificial structures—applicable to any generative or reconstruction network.

3. Latent-space organization for interpretable monitoring: Use a decoder-probe analysis to verify that latent dimensions are physically organized (e.g., pooled positions map to relaxation-time ranges, channels refine resolution). This enables unsupervised, interpretable condition monitoring via Euclidean distances in latent space, directly correlating with known operational events.

4. Dataset-agnostic generalization via fixed architecture: Train once on a broad synthetic/physics-informed basis and apply the same network configuration across multiple experimental datasets (SOFC, SOEC, multi-regime) without retuning, demonstrating transferability across operating conditions and cell types.

5. Implicit inductive regularization for inductive artifacts: Introduce a trainable parameter for inductive contributions and a large penalty term (λL = 106) to suppress non-physical inductive artifacts, improving robustness in real EIS data where such contributions are common.

6. Unified pipeline replacing manual preprocessing: Replace multi-step, hand-tuned preprocessing (regularization selection, peak fitting, etc.) with a single end-to-end model that outputs DRT, series resistance, and latent health indicators simultaneously, reducing human bias and improving reproducibility.

What the Improved AI System Can Do:

• Reconstruct physically consistent distributions of relaxation times from impedance spectra with <1.1% range-normalized RMSE across diverse experimental datasets, without per-dataset tuning.

• Resolve overlapping relaxation processes even when the Nyquist plot shows a single broad arc (e.g., peak-position errors of 0.065 and 0.025 decades for one-decade separation).

• Detect and localize operational events (fuel starvation, shutdowns, power outages, facility relocation) in real time via latent-space distance metrics, with peaks coinciding exactly with known events.

• Track multi-phase degradation trajectories (co-electrolysis, recovery, reversible cycling) using the same latent distance metric, reproducing phase-by-phase trends from prior frequency-resolved analyses.

• Provide interpretable latent representations where each dimension has a physical meaning (relaxation-time region and resolution), enabling users to understand why a deviation is flagged.

• Generalize across different electrochemical systems (fuel cells, electrolyzers) and operating regimes (steady-state, pulsating, reversible) with zero architectural changes, making it suitable for large-scale automated diagnostics in energy research and industrial monitoring.

Estimating the distribution of relaxation times (DRT) fromelectrochemical impedance spectroscopy (EIS) is an ill-posed inverse problem that is highly sensitive to regularisation choices. We propose a physics-informed convolutional autoencoder that estimates DRT directly from EIS data without spectrum-specific tuning. A discretised relation between impedance and the DRT is embedded in the training process, constraining the network to produce impedance-consistent distributions. The model resolves overlapping relaxation processes in synthetic two-ZARC spectra and accurately reconstructs measurements from three independent solid oxide fuel and electrolysis cell datasets, with range-normalised errors below 1.1%. Decoder-probe analysis shows that the learned latent representation is organised according to relaxation timescale. Distances in this latent space capture operating changes, hydrogen-shortage events, and long-term degradation. The same lightweight architecture is applied across all datasets without modification, providing consistent DRT estimation and an interpretable basis for condition monitoring.