Neural posterior estimation for scalable and accurate inverse parameter inference in Li-ion batteries

Malik Hassanaly; Corey R. Randall; Peter J. Weddle; Paul J. Gasper; Conlain Kelly; Tanvir R. Tanim; Kandler Smith

physics.data-anApr 2, 2026

Neural posterior estimation for scalable and accurate inverse parameter inference in Li-ion batteries

Authors: Malik Hassanaly, Corey R. Randall, Peter J. Weddle, Paul J. Gasper, Conlain Kelly, Tanvir R. Tanim, Kandler Smith

Organizations: Computational Science Center, National Laboratory of the Rockies, Golden, CO 80401 · Energy Conversion and Storage Systems Center, National Laboratory of the Rockies, Golden, CO 80401 · Energy Storage Research and Analysis Department, Idaho National Laboratory, Idaho Falls, ID 83415

Abstract

Diagnosing the internal state of Li-ion batteries is critical for battery research, operation of real-world systems, and prognostic evaluation of remaining lifetime. By using physics-based models to perform probabilistic parameter estimation via Bayesian calibration, diagnostics can account for the uncertainty due to model fitness, data noise, and the observability of any given parameter. However, Bayesian calibration in Li-ion batteries using electrochemical data is computationally intensive even when using a fast surrogate in place of physics-based models, requiring many thousands of model evaluations. A fully amortized alternative is neural posterior estimation (NPE). NPE shifts the computational burden from the parameter estimation step to data generation and model training, reducing the parameter estimation time from minutes to milliseconds, enabling real-time applications. The present work shows that NPE calibrates parameters equally or more accurately than Bayesian calibration, and we demonstrate that the higher computational costs for data generation are tractable even in high-dimensional cases (ranging from 6 to 27 estimated parameters), but the NPE method can lead to higher voltage prediction errors. The NPE method also offers several interpretability advantages over Bayesian calibration, such as local parameter sensitivity to specific regions of the voltage curve. The NPE method is demonstrated using an experimental fast charge dataset, with parameter estimates validated against measurements of loss of lithium inventory and loss of active material. The implementation is made available in a companion repository (https://github.com/NatLabRockies/BatFIT).

Figures & tables

Scaled parameter	Discharge ranges	Charge Ranges
Anode exchange current density $i_{0,\rm{a}}^{\prime}$	[0.1, 4.0]	Same as discharge
Solid cathode diffusivity $D_{\rm s,c}^{\prime}$	[0.2, 10.0]	Same as discharge
Initial anode intercalation fraction $x_{0,\rm{a}}^{\prime}$	[0.6, 1.077]	[0.3, 2.0]
Initial cathode intercalation fraction $x_{0,\rm{c}}^{\prime}$	[0.88, 1.6]	[0.78, 1.08]
Cathode exchange current density $i_{0,\rm{c}}^{\prime}$	[0.1, 1.6]	Same as discharge
Solid cathode active material volume fraction $\varepsilon_{\rm{s,c,AM}}^{\prime}$	[0.7, 1.0]	Same as discharge

Table 1: Ranges of the uniform priors used for the parameters calibrated with the Comparison dataset and the XCEL dataset. The parameters calibrated are unitless efficiency factors.

Figure 1: Schematic illustration of the neural net architectures used for CNPE (left) and the model surrogate (right). “CNN" refers to convolutional layers, and “FCNN" refers to fully connected layers.

Figure 2: Training results of the discharge surrogate. Left: train ( ) and test ( ) losses history. Right: discharge voltage curves obtained with different parameters

\theta

with the SPM ( ) and the surrogate ( ).

Figure 3: Train ( ) and test ( ) losses history for CNPE trained on the discharge comparison dataset. NLL denotes the batch-averaged negative log-likelihood.

Metric	Bayesian calibration		CNPE
	Discharge $\rm{C}/2$	Charge $4\rm{C}$	Discharge $\rm{C}/2$	Charge $4\rm{C}$
Parameter error (mean) $\varepsilon_{\rm PE}$	$8.1\%$	$6.14\%$	$7.68\%$	$4.41\%$
Parameter error (sample) $\varepsilon_{\rm PE,s}$	$9.98\%$	$7.33\%$	$10.58\%$	$6.32\%$
Coverage error $\varepsilon_{\rm CE}$	$62.45\%$	$70.45\%$	$9.91\%$	$1.68\%$
Voltage error (surrogate) $\varepsilon_{\rm VE,Surr}$	$0.801~{\rm{mV}}$	$0.792~{\rm{mV}}$	$3.096$ mV ( $2.328$ mV)	$5.493$ mV ( $2.862$ mV)
Voltage error (physics-model) $\varepsilon_{\rm VE,Phy}$	$1.751{\rm{mV}}$	$0.919~{\rm{mV}}$	$4.624$ mV ( $3.707$ mV)	$5.773$ mV ( $2.598$ mV)

Table 2: Accuracy metric comparisons between CNPE and Bayesian calibration on the validation set of the Comparison dataset. Best results are highlighted in bold. For CNPE, median voltage errors are reported in parentheses.

Figure 4: Left: example of near-matching posteriors between Bayesian calibration ( ) and CNPE ( ). Right: example of posterior mismatch between Bayesian calibration ( ) and CNPE ( ). Bayesian calibration done with a more accurate surrogate is also shown ( ). Solid lines ( ) denote the true values.

Parameter	Discharge $\rm{C}/2$	Charge $4\rm{C}$
Joint	$34.20\%$	$27.50\%$
$i_{0,\rm{a}}^{\prime}$	$101.33\%$	$62.14\%$
$D_{\rm s,c}^{\prime}$	$109.01\%$	$57.06\%$
$x_{0,\rm{a}}^{\prime}$	$1.72\%$	$2.77\%$
$x_{0,\rm{c}}^{\prime}$	$1.21\%$	$4.49\%$
$i_{0,\rm{c}}^{\prime}$	$32.18\%$	$24.23\%$

Table 3: Wasserstein distance

\varepsilon_{\rm W}

computed for the joint and the marginal posteriors. Parameters involved in the LLI and LAM calculation are indicated in bold.

Figure 5: Conditional average of the relative error for each calibrated parameter evaluated on the discharge dataset ( ) and the charge dataset ( ).

Figure 6: For the Comparison charge and discharge test datasets. Left: average relative errors for each parameter inferred. Right: average predicted parameter standard deviation.

Panel of Figure 7 — Figure 7: Absolute SHAP value obtained for 1000 discharge (left) and charge (right) realizations of the Comparison dataset, where the target is the mean value of $\varepsilon_{\rm s,c,AM}$ . The darker, the higher the absolute SHAP value, i.e the more informative for $\varepsilon_{\rm s,c,AM}$ .

Figure 8: Schematic illustration of the modified architecture for CNPE. “CNN” refers to convolutional layers, “FCNN” refers to fully connected layers, and “Cat” refers to a concatenation operation.

Figure 9: LLI (left) and

\rm{LAM}_{\rm PE}

(right) measured during cycle aging ( ) and predicted mean ( ) and standard deviation ( ). For the loss of lithium inventory (left),

\rm{LLI}_{\rm ch}

( ) and

\rm{LLI}_{\rm dis}

( ) are shown.

Figure 10: LLI (left) and

\rm{LAM}_{\rm PE}

(right) measured during cycle aging ( ) and predicted mean ( ) and standard deviation ( ), with Li conservation enforced through the prior. For the loss of lithium inventory (left),

\rm{LLI}_{\rm ch}

( ) and

\rm{LLI}_{\rm dis}

( ) are shown.

Figure 11: Left: performance metric loss with respect to the training data size for the 6-dimensional, 16-dimensional and 27-dimensional cases. Right: number of training data point required to lose at most 5% for the performance metric. Results at 6,16 and 27 dimensions ( ) are super-imposed with a linear trend ( ) and associated multiplicative uncertainty ( ).

Appendix figures & tables6 assets

Supplementary material from the paper’s appendix.

Appendix

Parameter [unit]	Value
Nominal capacity [Ah]	$1.89\times 10^{-2}$
Cell area [m 2 ]	$1.4\times 10^{-3}$
Initial $\text{Li}{\vphantom{\text{X}}}^{\text{+}}$ concentration [kmol m -3 ]	1.2
Anode thickness [ $\mu$ m]	44
Anode secondary particle radius [ $\mu$ m]	4
Anode solid-phase volume fraction [-]	0.6

Appendix

Table 4: Reference parameters values. Those are the values scaled with the efficiency factors inferred.

Hyper-parameter	Optimal	Set of values explored
Maximal number of neurons per hidden layer	2048	$\{512,1024,2048\}$
Number of hidden layers	3	$\{3,5,9\}$
Initial learning rate	$10^{-3}$	$\{10^{-3},10^{-2}\}$
Batch size	$2^{16}$	$\{2^{16},2^{17}\}$

Appendix

Table 5: Hyperparameters of the surrogate model and associated search space during hyperparameter optimization.

Hyper-parameter	Optimal	Set of values explored
Number of convolutional layer	2	$\{2,4,6\}$
Number of filters per convolutional layer	512	$\{512,1024,2048\}$
Number of fully connected hidden layers	4	$\{1,2,4\}$
Number of neurons per hidden layers	2048	$\{512,1024,2048\}$
Initial learning rate	$10^{-4}$	$\{10^{-4},10^{-5}\}$
Batch size	$2^{8}$	$\{2^{8},2^{9}\}$

Appendix

Table 6: Hyperparameters of the CNPE model and associated search space during hyperparameter optimization.

Metric	Bayesian calibration		CNPE
	Discharge $\rm{C}/2$	Charge $4\rm{C}$	Discharge $\rm{C}/2$	Charge $4\rm{C}$
$\varepsilon_{\rm PE}$	$8.89\%$	$7.29\%$	$8.88\%$	$5.96\%$
$\varepsilon_{\rm PE,s}$	$11.52\%$	$9.36\%$	$12.18\%$	$8.57\%$
$\varepsilon_{\rm CE}$	$22.78\%$	$13.45\%$	$4.22\%$	$0.68\%$
$\varepsilon_{\rm VE,Surr}$	$3.060{\rm{mV}}$	$2.927{\rm{mV}}$	$5.324\rm{mV}(4.906\rm{mV})$	$10.541\rm{mV}(6.913\rm{mV})$
$\varepsilon_{\rm VE,Phy}$	$4.308{\rm{mV}}$	$3.016{\rm{mV}}$	$6.670\rm{mV}(5.962\rm{mV})$	$9.802\rm{mV}(6.676\rm{mV})$

Appendix

Table 7: Accuracy metric comparisons between CNPE and Bayesian calibration on the validation set of the Comparison dataset. Best results are highlighted in bold. For CNPE, median voltage errors are reported in parentheses. The noise model is

\mathcal{U}(-5.76\rm{mV},5.76\rm{mV})

Metric	Bayesian calibration		CNPE
	Discharge $\rm{C}/2$	Charge $4\rm{C}$	Discharge $\rm{C}/2$	Charge $4\rm{C}$
$\varepsilon_{\rm PE}$	$9.69\%$	$8.48\%$	$9.88\%$	$7.46\%$
$\varepsilon_{\rm PE,s}$	$12.87\%$	$11.15\%$	$13.08\%$	$10.48\%$
$\varepsilon_{\rm CE}$	$10.12\%$	$6.45\%$	$4.55\%$	$4.45\%$
$\varepsilon_{\rm VE,Surr}$	$5.843{\rm{mV}}$	$5.842{\rm{mV}}$	$8.025{\rm{mV}}(7.716{\rm{mV}})$	$14.358\rm{mV}(11.460\rm{mV})$
$\varepsilon_{\rm VE,Phy}$	$6.594{\rm{mV}}$	$5.941{\rm{mV}}$	$9.108{\rm{mV}}(8.435{\rm{mV}})$	$13.513{\rm{mV}}(10.860{\rm{mV}})$

Appendix

Table 8: Accuracy metric comparisons between CNPE and Bayesian calibration on the validation set of the Comparison dataset. Best results are highlighted in bold. For CNPE, median voltage errors are reported in parentheses. The noise model is

\mathcal{U}(-11.52\rm{mV},11.52\rm{mV})

Scaled parameter	Ranges
Solid anode active material volume fraction $\varepsilon_{\rm{s,a,AM}}^{\prime}$	[0.7, 1.0]
Solid anode diffusivity $D_{\rm s,a}^{\prime}$	[0.2, 2.0]
Initial electrolyte $\text{Li}{\vphantom{\text{X}}}^{\text{+}}$ concentration $c_{\rm e,0}^{\prime}$	[0.5, 2.0]
Electrolyte volume fraction in the anode $\varepsilon_{\rm{e,a}}^{\prime}$	[0.7, 1.0]
Electrolyte volume fraction in the cathode $\varepsilon_{\rm{e,c}}^{\prime}$	[0.7, 1.0]
Electrolyte volume fraction in the separator $\varepsilon_{\rm{e,s}}^{\prime}$	[0.7, 1.0]

Appendix

Table 9: Ranges of the uniform priors used for the high-dimensional parameters inference in Sec. 3.3 .

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Parameter Estimation Lithium-Ion Batteries Physics-Informed ML Neural Surrogate Modeling

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Accurate, real-time, yet non-destructive estimation of internal states in lithium-ion batteries is critical for predicting degradation, optimizing usage strategies, and extending operational lifespan. Here, we introduce PINEAPPLE (Physics-Informed Neuro-Evolution Algorithm for Prognostic Parameter inference in Lithium-ion battery Electrodes), a novel framework that integrates physics-informed neural networks (PINNs) with an evolutionary search algorithm to enable rapid, scalable, and interpretable parameter inference with potential for application to next-generation batteries. The meta-learned PINN utilizes fundamental physics principles to achieve accurate zero-shot prediction of electrode behavior with test errors below 0.1

\%

while maintaining an order-of-magnitude speed-up over conventional solvers. PINEAPPLE demonstrates robust parameter inference solely from voltage-time discharge curves across multiple batteries from the open-source CALCE repository, recovering the evolution of key internal state parameters such as Li-ion diffusion coefficients across usage cycles. Notably, the inferred cycle-dependent evolution of these parameters exhibit consistent trends across different batteries without any customized degradation physics-embedded heuristic, highlighting the effective regularizing effect and robustness that can be conferred through incorporation of fundamental physics in PINEAPPLE. By enabling computationally efficient, real-time parameter estimation, PINEAPPLE offers a promising route towards the non-destructive, physics-based characterization of inter-cell and intra-cell variability of battery modules and battery packs, thereby unlocking new opportunities for downstream on-the-fly needs in next-generation battery management systems such as individual cell-scale state-of-health diagnostics.

Abstract

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