PINEAPPLE: Physics-Informed Neuro-Evolution Algorithm for Prognostic Parameter Inference in Lithium-Ion Battery Electrodes
Authors: Karkulali Pugalenthi, Jian Cheng Wong, Qizheng Yang, Pao-Hsiung Chiu, My Ha Dao, Nagarajan Raghavan, Chinchun Ooi
Organizations: Singapore Institute of Manufacturing Technology, Agency for Science, Technology and Research, 5 Cleantech Loop, Singapore, 636732, Singapore · Institute of High Performance Computing, Agency for Science, Technology and Research, 1 Fusionopolis Way, Singapore, 138632, Singapore · Department of Mathematics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong SAR, China · Technology Centre for Offshore and Marine, Singapore, 12 Prince George’s Park, Singapore, 118411, Singapore · Engineering Product Design Pillar, Singapore University of Technology and Design, 8 Somapah Road, Singapore, 487372, Singapore · Centre for Frontier AI Research, Agency for Science, Technology and Research, 1 Fusionopolis Way, Singapore, 138632, Singapore
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.
Figures & tables
Figure 1: Discharge voltage-time (V-t) curves for the four selected CX2 batteries. A red-to-blue gradient color scheme is used to distinguish V-t curves across different cycles (CALCE CX2-34: 50–1726 cycles, CX2-36: 53–1958 cycles, CX2-37: 53–1274 cycles, CX2-38: 53–1949 cycles) during the battery’s degradation process, with the progression in cycles indicated by a change in color from blue to green to orange and finally red.
Figure 2: PINEAPPLE schematic. The nonlinear hidden layers of LE-PINN are meta-learned (pre-trained) through Baldwinian neuroevolution, which takes less than 600 seconds on a sparse simulation dataset with diffusion coefficients ( Dk ) spanning 3 orders of magnitude. The fine-tuning (forward solve) of the final layer for positive and negative electrodes together costs less than 20 milliseconds for specific SPM parameters. The evolutionary search for best fit cycle-dependent parameters to match the observed V-t curve, Vobs , can be completed in a few seconds. Representative examples for early-, mid-, and late-cycle experimentally observed V-t curves (obtained from CALCE CX2-37 Cycle 53, Cycle 638, and Cycle 1181 respectively) and the corresponding prediction from the PINEAPPLE framework are illustrated in the top portion of the Figure.
Figure 3: Predictive performance of LE-PINN model: examples of the solution vs. ground truth for unseen (top: out-of-sample, bottom: out-of-distribution) SPM parameters for positive and negative electrodes. The concentrations Cp and Cn are normalized by their respective initial values.
SPM parameter
Scaling factor
Relative error (train)
Relative error (test)
Evaluation time 1
Positive electrode 2 , Cp
Dp∈[3.9e−15,3.9e−13]Gp∈[1.01,4.03]
ηDp∈[1e−1,1e1]ηGp∈[1,4]
3.57e−4±2.99e−4
2.78e−4±2.61e−4
5.62 ms ±4.43 ms
Negative electrode 3 , Cn
Dp∈[3.9e−16,3.9e−13]
ηDn∈[1e−2,1e1]
7.94e−4±6.29e−4
1.29e−3±6.96e−4
5.64 ms ±4.38 ms
Table 1: Predictive performance of LE-PINN model.
Figure 4: Comparison of relative error and computational time between LE-PINN (obtained via fine-tuning of the final layer) and numerical solver from PyBaMM at different mesh resolutions, using an out-of-sample SPM parameter for the positive electrode as a representative example. LE-PINN achieves comparable accuracy to that of the numerical solution with a mesh size of 32, while requiring significantly less (10 × ) computational time. For all cases, errors are computed with respect to the reference numerical solution generated by PyBaMM using a mesh size of 1024.
Figure 5: Simulated discharge V-t curves (dashed) using different SPM and terminal voltage calculation parameters. These curves are used for a preliminary evaluation before conducting inverse inference on the real-world dataset. The simulated curves correspond well qualitatively to the early (blue), middle (green), and late (red) cycles of the battery degradation process. The range of measured V-t curves, from the real dataset as described in Section 2 , is shown in the background as semi-transparent solid lines.
Figure 6: Validation of the PINEAPPLE framework on three synthetic V-t curves, representing the early (blue), middle (green), and late (red) stages of the battery degradation process. The first row of plots compares the synthetic V-t curves with LE-PINN-predicted V-t curves, based on inferred parameters obtained from 100 optimization runs with different random initializations. The second row illustrates the correlation between inference error (i.e., deviation of inferred scaling factors from ground truth) and MSE (difference between synthetic and LE-PINN-predicted V-t curves). Data points with MSE values below the 50th percentile threshold are highlighted in dark purple. Additionally, Spearman’s rank correlation coefficients for varying MSE percentile thresholds are provided. The third row presents the distribution of inferred scaling factors, excluding the worst 50 % of MSE cases (which suggest poorer convergence during the evolutionary search). The dotted lines indicate their respective ground truth values. The final row overlays the synthetic V-t curves with the LE-PINN-predicted V-t curves after filtering out the highest 50 % MSE cases.
Figure 7: Distribution of inferred parameter values and MSE between measured and predicted V-t curves over cycles during the degradation process for four CX2 batteries. Each distribution is aggregated from 20 independent runs of evolutionary search with different random initializations, for the 50 % of cases with the lowest MSE.
Figure 8: Sensitivity of terminal voltage (V–t) curves to local perturbations in SPM scaling factors. The figure shows the response of the terminal voltage to one-at-a-time perturbations in the scaling factors ηDp , ηGp , and ηDn under two representative reference states. Left column: reference case with ηDp =0.5, ηGp =3.5, and ηDn =0.1. Right column: reference case with identical positive electrode parameters and a reduced negative electrode diffusion coefficient ηDn =0.05. In each subplot, the black solid curve denotes the reference V-t trajectory, while dotted curves correspond to small positive and negative perturbations of a single scaling factor, with all other parameters held fixed. To provide physical context, the range of measured V-t curves from the real dataset, as described in Section 2 , is shown in the background as semi-transparent solid lines.
Figure 9: Scatter plots showing the relationship between inferred electrochemical parameters ( ηcmax,p , ηDn , ηDp , ηGp ) and discharge capacity for four batteries (CX2-34, CX2-36, CX2-37, CX2-38). The overlaid trend lines are manually drawn to guide qualitative interpretation of parameter evolution and do not represent fitted regression models.
Appendix figures & tables1 asset
Supplementary material from the paper’s appendix.
Appendix
Parameter
Symbol
Scaling factor 1 , η
Reference value
Lithium Cobalt Oxide (LCO, Cathode)
Diffusion coefficient [m2/s]
Dp
1e−1−1e1
3.9e−14
[ 22 ]
Geometrical composite coefficient
Gp
1−4
3ϵpApLpRp
Maximum Li concentration [mol/m3]
cmax,p
0.8−1.2
51000
[ 28 ]
Initial Li concentration [mol/m3]
Cp
0.82
30730
[ 28 ]
Particle radius [m]
Rp
5
3.0e−6
[ 28 ]
Appendix
Table 2: SPM and terminal voltage parameter values.
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).
Malik Hassanaly, Corey R. Randall, Peter J. Weddle +4
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
Physics-informed neural networks (PINNs) have emerged as a powerful tool for solving nonlinear partial differential equations (PDEs), including battery electrochemical models. They typically en-force conservation laws within the loss function to ensure physically consistent solutions. Tradi-tional numerical methods such as finite difference, finite volume, and finite element techniques, re-ly on discretization and can be computationally expensive for nonlinear systems. To address this challenge, PINNs offer improved scalability, particularly for reduced-order models like the single particle model with electrolyte (SPMe). The SPMe describes lithium-ion battery dynamics through coupled diffusion, transport, reaction kinetics, and voltage equations. Despite these advantages, training SPMe-based PINNs from scratch for different battery chemistries or operating conditions is demanding and often leads to slow convergence. To overcome this limitation, this work introduces a transfer learning framework for SPMe-PINNs. The model is first pretrained to learn general elec-trochemical dynamics and then adapted to a target battery by transferring weights, freezing se-lected layers, and fine tuning the remaining parameters, including estimating key electrochemical variables. Validation using PyBaMM demonstrates accurate voltage prediction, indicating that the proposed approach preserves electrochemical consistency while reducing training time and ena-bling efficient generalization across batteries.
Gift Modekwe, Qiugang Lu
Department of Chemical Engineering, Texas Tech University, Lubbock, TX 79409, USA
Accurate prediction of lithium-ion battery state of health (SOH) is essential for reliable energy storage operation. However, purely data-driven models may generalize poorly across cycling protocols and produce physically implausible behavior during long-term extrapolation. We developed a physics-informed differentiable degradation modeling framework (PiDDM) for battery SOH prediction. PiDDM incorporates empirical Arrhenius degradation kinetics associated with solid electrolyte interphase growth and loss of lithium inventory into the training objective, encouraging physically consistent capacity fade under diverse operating conditions. The framework was evaluated using a public dataset of 55 batteries cycled under six operating protocols. PiDDM achieved the lowest average prediction error among the evaluated models and substantially reduced mean squared error relative to a multilayer perceptron and a baseline physics-informed neural network. For extrapolation, the models were trained on the first 90% of each battery's cycle life and evaluated on the unseen final 10%. PiDDM captured accelerated end-of-life degradation while avoiding the nonphysical capacity regeneration produced by the baseline models. These results show that incorporating degradation physics into neural network training improves predictive accuracy and physical consistency, providing a promising approach for practical battery health monitoring.
Zeping Chen, Ruda Jian, Sachin Sigdel +3
Department of Chemical and Biomolecular Engineering, University of Notre Dame, Notre Dame, IN, USA · Department of Mechanical Engineering, The University of Texas at Dallas, Richardson, TX, USA · Department of Aerospace and Mechanical Engineering, University of Notre Dame, Notre Dame, IN, USA +1