The Deep-Prior Distribution of Relaxation Times

Our investigation started with an illustrative example. We drew the EIS of a single ZARC element with ${\sigma }_{n}=0.1\,{{\rm{\Omega }}}^{\displaystyle \frac{1}{2}}$ as shown in Fig. 2a. Again, the reader can find the analytical formulas and the parameters used in Table I and Table II, respectively. Then, we minimized the loss ${\mathcal L} \left({\boldsymbol{\theta }}\right),$ see (4). The loss, which is plotted as a function of the iteration number (iter) in Fig. 2c, decreases with increasing iteration number and so does the error defined as

$$\begin{eqnarray}&&{\rm{error}}\left({\rm{iter}}\right)={\parallel {\boldsymbol{\gamma }}({\rm{\zeta }},{\theta }_{{\rm{iter}}})-{\gamma }_{{\rm{exact}}}\parallel }_{2}\end{eqnarray} \tag{ 6 }$$

and shown in Fig. 2d. Two vertical dashed lines are plotted in Fig. 2d and correspond to the early stopping and minimum error or optimal iteration. We also compared the $\gamma \left(\mathrm{ln}\,\tau \right)$'s obtained at those two iterations, see Fig. 2b. Both early-stopping and optimal values are capable of closely matching the EIS and the exact DRT, where only small ridges are present at the base of the $\gamma \left(\mathrm{ln}\,\tau \right),$ i.e., for 10⁻² $\leqslant \,\tau /s\,\leqslant$ ¹0⁻¹ and ¹0¹ $\leqslant \tau /s\,\leqslant$ 10².

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We then tested if the method can recover the DRT and EIS of two partially overlapping ZARC elements. The Nyquist plots of the two EIS spectra can be found in Figs. 3a and 3c with the parameters from Table II. The DP-DRT recovers closely the exact EIS. The $\gamma \left(\mathrm{ln}\,\tau \right)$ can be obtained with a similarly limited discrepancy, as shown in Figs. 3b and 3d.

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As proposed by Ulyanov and co-workers,⁶⁴ adding a regularizing term to the loss can enhance images obtained by the DIP. In that regard, Dittmer and co-workers⁶⁸ suggested to output $\lambda$ from the network itself, where the optimal value is obtained by applying L-curve-like criterion. We attempted to implement a variation of what was proposed in these two articles and added to the loss an approximation of $\gamma \left(\mathrm{ln}\,\tau \right)$'s 2nd derivative.^19,56,62,63 Furthermore, we set $\lambda \,\geqslant$ 10⁻⁶. We note that early stopping was still used. As shown in Fig. 4, this procedure leads only to a slight visual improvement in the recovered $\gamma$ and ${\boldsymbol{Z}}$ because the regularizing parameter monotonically decreased to 10⁻⁶ with the number of iterations. A similar improvement cannot be seen if the lower positive bound on $\lambda$ is not present (i.e. $\lambda \,\geqslant$ 0) as $\lambda \to 0$ with ${\rm{iter}}\to \infty .$

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One may wonder what happens to the DP-DRT method if the experimental noise intensifies. We have tested this by increasing ${\sigma }_{n}$ by one order of magnitude to ${\sigma }_{n}=1\,{{\rm{\Omega }}}^{\tfrac{1}{2}}.$ This choice leads to heavily corrupted EIS spectra, as shown in Fig. 5a. The DP-DRT method can still recover the function $\gamma (\mathrm{ln}\,\tau )$ well but a few additional ridges are triggered by the extra noise. It is important to note that, as shown in Fig. 5b, early stopping corresponds to an iteration where the error is near its minimum. As the number of iterations increases, overfitting of ${\boldsymbol{Z}}$ is expected to occur, leading to an increased error on the latent ${\boldsymbol{\gamma }}.$ If we perform the same procedure with smoothing by using the DPλ-DRT with $\lambda \,\geqslant$ 0.1, the recovered DRT does not show any ridges, see Fig. 5d. However, both the loss and the error increase as shown in Fig. 5f. We must stress that a large $\lambda ,$ i.e., $\lambda \,\geqslant$ 0.1 was chosen in order to emphasize the impact of the regularizing term.

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Repeating the same inversion procedure numerous times can help assess the robustness of the approach, which was conducted by performing 1,000 stochastic experiments under the same hypotheses as above. These experiments were followed by DP-DRT inversion with early stopping. In Fig. 6a, we plot the mean DP-DRT and the 5 and 95% confidence bands. It can be observed that the DRT shape is well captured, albeit with some scatter, especially near the peak, indicating strong sensitivity to noise at that location. Further, Fig. 6b plots the probability distribution function (PDF) of the relative error defined as ${\rm{rel}}.\,{\rm{error}}({\rm{iter}})={\parallel \gamma (\zeta ,{{\boldsymbol{\theta }}}_{{\rm{iter}}})-{\gamma }_{{\rm{exact}}}\parallel }_{2}/{\parallel {\gamma }_{{\rm{exact}}}\parallel }_{2}.$ Consistent with the data shown in Fig. 6a, Fig. 6c indicates that the relative error has a broad distribution. For comparison, Fig. 6b shows the outcome of the DPλ-DRT deconvolution of the same stochastic experiments with the DRT appearing to be smoother. In contrast with the outcome of the DP-DRT (without $\lambda$), the recovered $\gamma \left(\mathrm{ln}\,\tau \right)$ is insensitive to the noise, showing a narrow distribution for the relative error, see Fig. 6d. This is consistent with intuition: because of the large additional penalty of $\lambda \geqslant 0.1$, the bias is large, but the variance is small.

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We also tested whether the DP-DRT method can be used to fit non-conventional elements, whose DRT displays discontinuities and infinite values. For this reason, the EIS was computed for three circuits corresponding to the Havriliak-Negami (HN),⁷⁸ piecewise constant (PWC),⁶² and fractal⁵⁶ models. Then, the 'exact' impedance was corrupted with noise according to (4) with ${\sigma }_{n}=0.1\,{{\rm{\Omega }}}^{\tfrac{1}{2}},$ see Fig. 7. As shown there, the DP-DRT is able to recover both $Z\left(f\right)$ and $\gamma \left(\mathrm{ln}\,\tau \right)$ well. This is evident for the HN and PWC elements, see Figs. 7a–7d. For the fractal element, whose impedance and DRT are shown in Figs. 7e and 7f, respectively, oscillations are visible for $\tau \,\leqslant$ 1 s, though the timescale corresponding to the discontinuity appears to be captured well. We carried out identical stochastic experiments for a higher noise level by increasing ${\sigma }_{n}$ to ${\rm{0}}\mathrm{.5}\,{{\rm{\Omega }}}^{\displaystyle \frac{1}{2}}.$ As reported in Fig. 8, increasing the random experimental error does not affect the recovery of the EIS significantly, but more oscillations appear in the deconvolved $\gamma \left(\mathrm{ln}\,\tau \right).$ These simulation results are consistent with the analysis outlined above.

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We also studied the performance of the DP-DRT and DPλ-DRT models for sharp DRTs by choosing similar synthetic experiments as those reported in An introductory example where the key differences are the number of collocation point $N=$ 161 and $\phi$ either 0.9 or 0.99. For $\phi \,=$ 0.9, DP-DRT and DPλ-DRT models can match the exact DRT well, Fig. S3. If instead $\phi \,=\,$0.99, the timescale resolution is not sufficient to capture the peak value ( ∼500 ${\rm{\Omega }}$) as shown in Figs. S3d and S4d. Unsurprisingly, $\gamma \left(\mathrm{ln}\,\tau \right)$ obtained by DPλ-DRT deconvolution displays a broader peak width than that from the DP-DRT, see Fig. S5. This difference is due to the penalty used and can be overcome using lasso regularization.⁵⁶

We first analyzed an EIS spectrum obtained by testing a symmetrical solid oxide fuel cell (SOFC). For comparison, the EIS is repeated in the Nyquist plots shown in Figs. 9a–9c. 15% Samarium-doped ceria was used as the electrolyte and sandwiched between two electrodes consisting of a mixture of Ag and Ba_0.9La_0.1FeO_3−δ.⁷⁹ The data was collected at 650 °C in a mixture of N₂ and O₂ (${p}_{{O}_{2}}=$ 0.21 atm) in the frequency range from 0.1 Hz to 8.56 × 10⁴ Hz with 15 points per decade.

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We first modeled the experimental data with an equivalent circuit model (ECM) consisting of a ZARC element plus an inductance. The parameters estimated by regression are listed in Table III. The fitted EIS and obtained DRT are shown in Figs. S1a and S1b (available online at stacks.iop.org/JES/167/026506/mmedia), respectively. The EIS obtained by DP-DRT without regularization is shown in Fig. 9a. The corresponding $\gamma \left(\mathrm{ln}\,\tau \right)$ is plotted in Fig. 9d, where, for reference, the DRT of the ECM^54,60 is also shown. The DP-DRT method can recover well both the impedance and the DRT even without regularization. This is shown in Fig. 9g, where the loss (4) and the discrepancy are defined as

$$\begin{eqnarray}&&\mathrm{discrepancy}({\rm{iter}})={\parallel \gamma ({\boldsymbol{\zeta }},{\theta }_{{\bf{i}}{\bf{t}}{\bf{e}}{\bf{r}}})-{\gamma }_{{\bf{E}}{\bf{C}}{\bf{M}}}\parallel }_{2}\end{eqnarray} \tag{ 7 }$$

are plotted as a function of the iteration number. We further report the DPλ-DRT results obtained by setting $\lambda \,\geqslant$ 10⁻³, see Figs. 9b, 9e, and h, and $\lambda \,\geqslant \,$10⁻², see Figs. 9c, 9f, and 9i. Consistent with intuition, the small ridges present in the DP-DRT deconvolution could be smoothed out by including a regularizer. Unsurprisingly, the larger the lower bound on λ is, the more prominent the loss. We should note again that no exact DRT is available, and the discrepancy between DP-DRT and ECM-based distribution serves only as a reference. To compare the performance of different models, we also calculated the mean squared error (MSE) defined as

$$\begin{eqnarray}&&\mathrm{MSE}({\rm{iter}})={\parallel {Z}_{\exp }-{Z}_{{\rm{DRT}}}\parallel }_{2}^{2}\end{eqnarray} \tag{ 8 }$$

with the results shown in Table SI. The DP-DRT method performed best with the lowest MSE value. In contrast, the ECM had the largest MSE among all four models.

Table III.  Parameters of the ECM (a single ZARC element plus an inductance) used to fit the SOFC experiment and corresponding numerical values obtained by regression. The error of each parameter outputted by ZView is also shown.

Parameter	Regressed value	Error
${R}_{\infty }$	1.81 Ω	4.32 × 10-4 Ω
${R}_{{\rm{ct}}}$	0.26 Ω	5.56 × 10-4 Ω
${\tau }_{0}$	2.19 × 10-4 s	1.31 × 10-5 s
$\phi$	0.69	2.42 × 10-3
${L}_{0}$	5.36 × 10-7 H	1.37 × 10-9 H

We also tested the DP-DRT model using lithium-ion battery (LIB) data taken from our earlier works.^54,56,63 An ECM consisting of a series of 3 ZARCs and an inductance was used to model the experimental EIS.⁵⁴ The parameters of the ECM are listed in Table IV, and the corresponding EIS spectra and DRT are shown in Figs. S2a and S2b, respectively. It is important to note that, as shown in Fig. S2a, the real and imaginary parts of the spectra are in the 0.11 to 0.16 Ω and 0 to 0.016 Ω range, respectively. Since the simulations are carried out in a single-precision floating-point format, the data was scaled to unity to avoid underflow or loss of accuracy. The results were then scaled back to the original values. Similar to the SOFC experiment, the DP-DRT without regularization can fit well to the impedance data, see Fig. 10a. Regarding the DRTs for low enough $\tau$'s, the DP-DRT does not have obvious discrepancies with respect to the DRT reference obtained from the ECM, as the two leftmost peaks of the ECM-DRT can be recovered well, see Fig. 10d. For the rightmost peak, which corresponds to the lowest frequencies probed, and whose corresponding arc is incomplete (see Fig. 10a) the DP-DRT recovery is inconsistent with that obtained with the ECM-based DRT. However, we believe that this is reasonable, especially considering that the uncertainty on any estimate obtained from that part of the EIS spectrum is expected to be high. If the MSE is used as a metric for evaluating the quality of the model, we note that DP-DRT performs better than the ECM, see Table SI.

Table IV.  Parameters of the ECM (3 ZARC elements) used to fit the LIB experiment and corresponding numerical values as obtained by regression. The error of each parameter outputted by ZView is also shown.

Parameter	Regressed value	Error
${R}_{\infty }$	0.11 Ω	4.44 × 10-4 Ω
${R}_{{\rm{ct}},1}$	1.69 × 10-2 Ω	8.48 × 10-4 Ω
${R}_{{\rm{ct}},2}$	2.12 × 10-2 Ω	4.42 × 10-4 Ω
${R}_{{\rm{ct}},3}$	5.23 × 10-2 Ω	1.89 × 10-3 Ω
${\tau }_{1}$	2.34 × 10-3 s	1.23 × 10-3 s
${\tau }_{2}$	0.20 s	1.08 × 10-2 s
${\tau }_{3}$	75.30 s	2.33 s
${\phi }_{1}$	0.54	2.31 × 10-2
${\phi }_{2}$	0.94	7.78 × 10-3
${\phi }_{3}$	0.79	1.03 × 10-2
${L}_{0}$	7.61 × 10-7 H	3.64 × 10-8 H

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As conducted for the SOFC deconvolution of subsection 3.2.1, we computed the DPλ-DRT by setting $\lambda \geqslant$ 10⁻³ (Figs. 10b, 10e, and 10h) and $\lambda \geqslant \,$10⁻² (Figs. 10c, 10f, and 10i). Consistent with our earlier insight, the predicted DRT curves are smoother than without regularization. Further, the loss increases if regularization is included affecting the recovered impedance. We point the reader to Fig. 10c where at around $f\,\sim \,$11 Hz corresponding to the DPλ-DRT with $\lambda \,\sim \,$10⁻²; deviations in both EIS and DRT can be observed. For the largest $\lambda ,$ the loss and MSE are also the largest among the cases studied, see Fig. 10i and Table S1. It should again be stressed that the DRT obtained from the ECM is only a reference and the "exact" DRT is unknown.