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

Technical Summary: Physics-informed Distribution of Relaxation Times Estimation and Latent-space Condition Monitoring

Problem Statement
Estimating the Distribution of Relaxation Times (DRT) from Electrochemical Impedance Spectroscopy (EIS) data is an ill-posed inverse problem. The relationship between impedance $Z(j\omega)$ and the DRT function $g(\log \tau)$ is governed by an integral equation (Eq. 1) that is highly sensitive to measurement noise and regularisation choices. Traditional methods, such as Equivalent Circuit Models (ECMs), require specific prior assumptions that are difficult to define for fractional-order systems. While DRT offers a model-free, continuous representation of relaxation processes, extracting it requires regularisation. Existing data-driven approaches, such as Deep Prior networks, often train a separate model for each spectrum, leading to inconsistent implicit regularisation. This makes it difficult to distinguish genuine electrochemical changes from fitting artifacts. Furthermore, many latent-space approaches treat the DRT transformation as a separate pre-processing step or a black-box feature extractor, failing to embed the governing physical laws directly into the learning objective.

Methodology
The authors propose a physics-informed convolutional neural network (CNN) framework that estimates DRTs directly from raw EIS data (real and imaginary impedance components) without spectrum-specific tuning.

• Architecture: The model utilizes an encoder-decoder CNN structure. The input consists of the real and imaginary parts of the impedance spectrum treated as two channels. The network outputs the DRT function $g(\log \tau)$.
• Physics-Informed Constraint: Unlike standard autoencoders that reconstruct the input directly, this framework embeds the discretised form of the governing integral relation (Eq. 1) into the training loop. The predicted DRT is numerically integrated (using the trapezoidal rule) to reconstruct the impedance spectrum $\hat{Z}$. The loss function is the Mean Squared Error (MSE) between the measured impedance and the reconstructed impedance, plus an $L_2$ penalty on inductive contributions. This forces the network to learn a DRT that is physically consistent with the impedance data.
• Auxiliary Components: A separate branch predicts the series resistance $R_s$, and a trainable parameter represents the inductance $L$.
• Architectural Optimization: To prevent checkerboard artifacts common in transposed convolutions, the decoder uses upsampling layers followed by standard convolutions. A grid search determined an optimal architecture with approximately 12,000 parameters, a receptive field of 57, and a bottleneck size of 160. This balance prevents overfitting while resolving overlapping relaxation processes.
• Latent Space Analysis: After training, the encoder produces a compact latent representation. The authors perform "decoder probe" analysis by activating individual latent channels to map them to specific relaxation time regions, revealing a hierarchical organization of the latent space based on timescales.

Key Contributions

1. Unified Physics-Informed Framework: The paper introduces a single CNN architecture that learns a shared latent representation across an entire dataset, enforcing consistent regularisation. This contrasts with methods that fit independent models to each spectrum, thereby ensuring that differences in estimated DRTs reflect true electrochemical changes rather than fitting artifacts.
2. Direct DRT Estimation without Tuning: The model estimates DRTs directly from raw EIS data without the need for manual regularisation parameter selection, frequency-band segmentation, or equivalent circuit assumptions.
3. Interpretable Latent Space: The study demonstrates that the learned latent space spontaneously organizes according to relaxation timescales. Decoder probe analysis confirms that specific latent channels correspond to distinct regions of the relaxation time axis, providing a physically interpretable basis for monitoring.
4. Generalisation Across Diverse Conditions: The framework is validated on three independent datasets involving Solid Oxide Fuel Cells (SOFC) and Solid Oxide Electrolysis Cells (SOEC) with varying operating regimes, degradation mechanisms, and campaign durations (from 60 to 3600 hours). The same unmodified architecture was applied to all datasets.

Results

• Reconstruction Accuracy: The model achieved range-normalised reconstruction errors below 1.1% across all three datasets, confirming that the predicted DRTs satisfy the underlying electrochemical constraints.
• Resolution of Overlapping Processes: On synthetic two-ZARC spectra with analytically known DRTs, the model successfully resolved overlapping relaxation processes separated by as little as one decade in time constant, even in the presence of Gaussian noise ($\sigma = 0.5 \, \Omega$).
• Condition Monitoring:
  • In the SOFC stack monitoring campaign, Euclidean distances in the latent space ($D_{ref}(t)$) successfully identified major operational events, including fuel starvation, emergency shutdowns, and power losses, with peaks in the distance metric coinciding with known events.
  • In the operating condition study, latent activation changes aligned with transitions identified by frequency-resolved Kullback–Leibler (KL) divergence analysis in the original study, detecting diffusion losses and air-electrode changes.
  • In the multi-regime degradation campaign, the latent space trajectory captured the evolution of degradation phases (steam electrolysis, co-electrolysis, reversible operation) and matched the trends observed in phase-by-phase DRT diagnostics.
• Comparison with Existing Methods: Compared to Tikhonov regularisation and per-spectrum neural networks, the proposed method produced smoother, more continuous temporal evolution of DRTs, avoiding abrupt transitions caused by varying regularisation strengths.

Significance and Claims
The paper claims that the primary significance of this work lies not merely in reconstruction accuracy, but in the emergence of a physically consistent and interpretable latent representation from a single, unified model. By embedding the physical forward model into the training process, the framework eliminates the need for manual preprocessing pipelines (regularisation tuning, circuit selection) that have historically been bottlenecks in large-scale EIS diagnostics.

The authors assert that this approach provides a "lightweight" and "fully reproducible" solution capable of generalizing across fundamentally different cell types, operating regimes, and degradation mechanisms without dataset-specific redesign. The latent space serves as a direct byproduct for condition monitoring, offering a computationally efficient alternative to complex DRT comparison metrics like Unbalanced Optimal Transport (UOT) or KL divergence, while maintaining the ability to detect degradation and operational anomalies with high fidelity.