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Physics-Informed Machine Learning for Battery Degradation Diagnostics: A Comparison of State-of-the-Art Methods

Sina Navidi, Adam Thelen, Tingkai Li, Chao Hu

arXiv:2404.04429v1cs.CE

TL;DR

Battery component degradation is difficult to quantify without destructive analysis, and existing health indicators do not identify internal degradation modes. This paper compares four PIML approaches for late-life diagnostics using short-term experimental data and physics-based simulations; the approaches outperform purely data-driven baselines, especially for late-life extrapolation.

  • Problem

    Directly quantifying internal battery degradation modes generally requires destructive component analysis, while common health indicators do not identify those modes.

    Source: §1 · pp. 2–3

  • Method

    The paper compares PINN, co-kriging, delta learning, and data augmentation for late-life degradation diagnostics using early-life experimental data and half-cell-model simulations.

    Source: §1 · p. 4

  • Results

    The PIML approaches outperform purely data-driven baseline models, with the improvement especially notable in late-life degradation extrapolation.

    Source: §8 · p. 31

  • Takeaways & Limitations

    The comparison is intended to clarify method strengths, limitations, and factors relevant to choosing an approach for degradation diagnostics and prediction tasks.

    Source: §1 · p. 4

    Online prediction remains constrained by the cost and time required for model training and inference, despite computational-efficiency design choices in the methods studied.

    Source: §7.1.4 · p. 28

Abstract

from arXiv · show

Monitoring the health of lithium-ion batteries' internal components as they age is crucial for optimizing cell design and usage control strategies. However, quantifying component-level degradation typically involves aging many cells and destructively analyzing them throughout the aging test, limiting the scope of quantifiable degradation to the test conditions and duration. Fortunately, recent advances in physics-informed machine learning (PIML) for modeling and predicting the battery state of health demonstrate the feasibility of building models to predict the long-term degradation of a lithium-ion battery cell's major components using only short-term aging test data by leveraging physics. In this paper, we present four approaches for building physics-informed machine learning models and comprehensively compare them, considering accuracy, complexity, ease-of-implementation, and their ability to extrapolate to untested conditions. We delve into the details of each physics-informed machine learning method, providing insights specific to implementing them on small battery aging datasets. Our study utilizes long-term cycle aging data from 24 implantable-grade lithium-ion cells subjected to varying temperatures and C-rates over four years. This paper aims to facilitate the selection of an appropriate physics-informed machine learning method for predicting long-term degradation in lithium-ion batteries, using short-term aging data while also providing insights about when to choose which method for general predictive purposes.

1 Introduction

Battery health models can estimate capacity and resistance, but do not reveal the internal degradation modes that explain those changes. Because direct component-level quantification relies on destructive analysis, the paper compares PIML approaches for late-life diagnostics without late-life training data.

1 Introduction

  • Component-level degradation modes are needed to explain capacity loss and resistance increase, but destructive analysis of retired cells limits direct quantification.The modes discussed are loss of active material on the positive and negative electrodes (LAMPE and LAMNE) and loss of lithium inventory (LLI).Source: §1 · pp. 2–3
  • Existing battery health estimators predict capacity or resistance from measurements, while prior non-destructive degradation methods include curve-feature analysis, diagnostic algorithms, neural networks, and physical models.Physical-model approaches can be computationally intensive and difficult to connect to real-world aging based on standard aging-test data.Source: §1 · pp. 2–3
  • PIML combines physical principles with machine learning, and prior battery studies applied it to health prediction, capacity degradation, and electrochemical quantities.Prior approaches include architectures combining physics-based models and machine learning, PINNs, and data augmentation.Source: §1 · pp. 3–4
  • The study compares PINN, co-kriging, delta learning, and data augmentation for late-life degradation diagnostics without late-life aging data for training.It uses data from a long-term cycling experiment involving 24 implantable-grade lithium-ion cells and examines factors including late-life data availability and degradation-process complexity.Source: §1 · p. 4

2 Dataset Description

The dataset comprises implantable-grade LCO/graphite cells cycled under six operating conditions for nearly five years. Repeated reference performance tests provide capacity and half-cell-model data for analysis.

2 Dataset Description

  • The study tested 24 implantable-grade lithium-cobaltoxide/graphite cells in groups of four under six operating conditions.Table 1 summarizes the aging test conditions.Source: §2 · p. 5, Table 1 · p. 5
  • Cells were charged at C/3, held at the upper-voltage cutoff until current fell below C/50, and discharged at constant current to 3.4 V.Aging tests ran in temperature-controlled chambers for nearly five years.Source: §2 · p. 5
  • Reference performance tests occurred every two weeks during the first three months and every four weeks thereafter to assess capacity and obtain data for half-cell analysis and state-of-health estimation.Before each test, chamber temperature was adjusted to 40 °C.Source: §2 · p. 5

3 Half-Cell Model

The half-cell model reconstructs a pseudo-OCV curve to quantify battery degradation modes. Its adjustable electrode-mass and slippage parameters also support degradation simulations for model training.

3 Half-Cell Model

  • The half-cell model reconstructs a pseudo-OCV curve from positive- and negative-electrode half-cell curves to quantify LAMPE, LAMNE, and LLI.The full-cell OCV is modeled as the potential difference between the electrode OCV curves at low rates, where kinetic and thermal effects are minimal.Source: §3 · p. 5
  • The remaining active masses mp and mn set electrode-curve widths, while slippage parameters δp and δn set their relative horizontal positions.The reconstructed curve is restricted to the usable voltage window, 3.4 V to 4.075 V for reference performance tests.Source: §3 · p. 5, §3.1 · p. 7

3.1 Degradation Mode Quantification Using the Half-Cell Model

  • Manual fitting adjusts mp, mn, δp, and δn to match measured C/50 voltage curves, while changes in these parameters represent active-material loss and lithium-inventory-related shifts.Differential-voltage curve features can guide fitting, but manual fitting is not efficient or practical for online diagnostics without human intervention.Source: §3.1 · pp. 6–7
  • Simulated incremental-capacity curves distinguish degradation patterns: active-material loss changes peak magnitude and width, while LLI primarily reduces peak magnitude.Combining active-material loss with LLI further reduces peak magnitude, especially the larger peak around 3.8 V.Source: §3.1 · p. 7

3.2 Simulating Aging Data Using the Half-Cell Model

  • The half-cell model generates many simulated capacity-voltage and incremental-capacity curves across degradation-parameter combinations for machine-learning training.The simulations vary half-cell model parameters and produce corresponding Q(V) and dQ/dV(V) curves.Source: §3.2 · p. 7

4 Problem Definition

The problem is to estimate cell capacity and three degradation modes from incremental-capacity curves using physics-informed learning. Training combines early-life experimental data with half-cell simulations, then tests predictions on late-life experimental data.

4 Problem Definition

  • Existing degradation-diagnostic approaches may require expensive physical-model evaluation or long-term aging data, making deployment time- and cost-intensive.The study investigates lightweight PIML methods for diagnostics using early-life experimental and physics-based simulation data.Source: §4 · pp. 7–8
  • The diagnostic task estimates capacity and LAMPE, LAMNE, and LLI from measured incremental-capacity curves, avoiding manual feature selection.The paper describes a four-method comparison of physics-informed machine-learning approaches.Source: §4 · p. 9
  • Early-life experimental data and half-cell simulations train the models to predict capacity and degradation parameters, with late-life experimental data reserved for testing.The framework fits four half-cell parameters and simulates medium- and late-life data for inclusion in training.Source: §4 · p. 9

5 Physics-Informed Machine Learning (PIML) Techniques

The section presents four PIML techniques that use late-life simulation data to support extrapolation beyond early-life experimental data. It describes their distinct model architectures and how each incorporates simulation and experimental information.

5 Physics-Informed Machine Learning (PIML) Techniques

  • The four techniques—PINN, data augmentation, multi-fidelity co-kriging, and elastic-net delta learning—use late-life simulation data to support extrapolation beyond early-life experiments.The methods are compared in this study as approaches to battery degradation diagnostics.Source: §5 · p. 9

5.1 Physics-Informed Neural Network (PINN)

  • PINN predicts half-cell parameters from dQ/dV curves, then uses half-cell physics and parameter and peak-position losses to constrain health estimates.The network predicts half-cell parameters; the half-cell surrogate maps them to capacity and lithium inventory, while peak-position differences constrain active-mass estimates.Source: §5.1 · pp. 9–12

5.2 Data Augmentation

  • Data augmentation adds half-cell simulation samples across the health-parameter design space, including late-life degradation, to early-life experimental data.One described variant selects simulated samples with the highest degradation values for Q, mp, mn, and LII.Source: §5.2 · p. 13

5.3 Delta Learning

  • Delta learning trains a secondary model to correct primary-model errors; here, co-kriging uses a probabilistic formulation, whereas elastic-net correction is deterministic.The study implements these approaches with GPR and elastic net, respectively.Source: §5.3 · p. 13

5.3.1 Delta Learning with Kriging (a.k.a Multi-Fidelity Co-Kriging)

  • Co-kriging treats early-life experiments as high fidelity and lifetime-spanning simulations as low fidelity, modeling high fidelity as scaled low fidelity plus a discrepancy Gaussian process.The implementation trains one GPR on simulation data and another on the discrepancy between experimental data and the low-fidelity prediction.Source: §5.3.1 · p. 14

5.3.2 Delta Learning with Elastic Net

  • Elastic-net delta learning uses simulation-trained estimation and an early-life-data-trained corrector, whose predictions are combined for the final estimate.The corrector learns the estimation model’s prediction bias in the light-degradation region, which is then used to correct late-life outputs.Source: §5.3.2 · p. 15

6 Results

The section compares four PIML methods with data-driven baselines, then examines training-data size, late-life extrapolation, and model sensitivity. Across the reported comparisons, PIML methods improve on their baselines, with outcomes varying by parameter and approach.

6 Results

  • The comparison examines each PIML model against its data-driven baseline, with additional analyses of training-data size, extrapolation, and sensitivity.For PINN, sensitivity analysis varied the relative weights of three loss terms; for co-kriging, it varied the kernel functions.Source: §6 · pp. 15–16, §6.6 · p. 23

6.1 Training and Test Setup

  • Models used the initial 10 characterization points and were assessed via four-fold cross-validation across 24 cells, with one cell from each group held out per fold; errors covered Q, mp, mn, and LII.The normalized average test error across the four folds was computed using RMSPE for Q, mp, mn, and LII.Source: §6.1 · p. 16, §6.2 · p. 18

6.3 Overall Performance Comparison

  • All PIML methods improved error rates over their data-driven counterparts; PINN performed better than its matched neural-network baseline, while co-kriging improved over GPR across the four health parameters.The elastic net was the most accurate overall baseline; GPR was especially capable on capacity (Q) and lithium inventory (LII), while elastic net was most accurate for active-mass parameters (mp and mn).Source: §6.3 · pp. 18–20

6.4 Performance Under Different Training Sample Sizes

  • PIML performance generally improved as training data increased from 90 to 180 points, while gains beyond 180 points were less prominent or insignificant.The 90, 180, and 270 points correspond to 5, 10, and 15 characterization points per training cell, respectively; baseline kriging accuracy remained consistent across these sizes.Source: §6.4 · pp. 20, 22, Figure 9 · p. 21

6.5 Extrapolation Performance

  • As cell degradation increased, regular neural-network extrapolation error rose substantially relative to the physics-informed approach, particularly in medium and late life.The comparison used test cell G2C2 at C/24 and 55°C; the reported RMSE covers mp, mn, and LII.Source: §6.5 · p. 22, Figure 10 · p. 23
  • For test cell G3C3, PINN and co-kriging tracked fitted health-parameter trajectories better than their baselines, especially for active masses mp and mn.Improvements for Q and LII were less pronounced; the example cell was tested at C/3 and 37°C.Source: §6.5 · p. 23, Figure 11 · p. 24

6.6.1 PINN

  • Combining all three PINN loss terms generally lowered error; equal weights minimized errors for Q, mn, and LII, while mp favored greater weight on the first two terms.Using any single loss term alone tended to produce higher error rates.Source: Figure 12 · p. 24, Effect of Loss Terms · p. 24

6.6.2 Co-Kriging

  • Co-kriging with a Matern32 kernel in each kriging model had the lowest average error across health parameters, although the polynomial kernel performed better for mn.The authors relate Matern32’s moderate smoothness to its flexibility, while noting that mn’s stronger nonlinearities may suit the polynomial kernel better.Source: §6.6.2 · pp. 26–27, Figure 14 · p. 26

7 Discussion

The discussion compares PIML methods across implementation and modeling trade-offs, then gives method-selection guidance for battery degradation diagnostics. PINNs are favored for generalization and complex degradation trends, while other methods offer advantages in scalability, ease, or uncertainty quantification.

7.1.1 Model Flexibility

  • PINNs can represent complex physics directly, but delta learning and data augmentation may require synthetic simulation data.PINNs can be useful when physics-based training data is difficult to generate, though their explicit mappings are not always available.Source: §7.1.1 · pp. 27–28, §7.1.9 · p. 29

7.1.2 Data Requirements

  • PINNs can achieve high accuracy with limited data, whereas delta learning relies on high-fidelity data and augmentation depends on relevant, diverse synthetic samples.Shallow PINNs may need less data; delta learning uses a corrector model to refine low-fidelity estimates.Source: §7.1.2 · p. 28

7.1.3 Ease of Implementation

  • Delta learning and data augmentation are simpler to implement than PINNs, while co-kriging requires more hyperparameter tuning and kernel selection.PINNs require physics-informed loss design and consistent gradient computation.Source: §7.1.3 · p. 28

7.1.5 Scalability to Large Applications

  • Data augmentation is the most straightforward of the considered methods to scale from small to larger problems, though scaling may require resource, algorithm, or hyperparameter adjustments.All methods were designed with computational efficiency in mind for online estimation, where training and inference cost and time are challenges.Source: §7.1.5 · pp. 28–29, §7.1.4 · p. 28

7.1.6 Generalization Capability

  • PINNs generalize better to unseen data and operating conditions, while delta learning and data augmentation depend on training-data trends.PINNs incorporate physics constraints directly into the loss function.Source: §7.1.6 · p. 29

7.1.7 Interpretability

  • PINNs make physics constraints explicit and interpretable, but data augmentation may not reveal input-output relationships as directly.All considered PIML methods showed promising prediction results on training and test datasets.Source: §7.1.7 · p. 29, §7.1.8 · p. 29

7.1.9 Applicability to Different Problem Types

  • PINNs can be difficult to apply without explicit physics mappings, co-kriging needs definable fidelity correlations, and data augmentation is versatile across fields.Data augmentation uses synthetic physics-model data, while delta learning can adapt through its choice of regression model.Source: §7.1.9 · p. 29, §7.2.3 · p. 31

7.1.10 Uncertainty Quantification Capability

  • PINNs require added methods such as ensembles or Bayesian neural networks for uncertainty estimates, whereas co-kriging provides prediction variances naturally.Bayesian or bootstrap methods can also add uncertainty estimates to delta learning and data augmentation with elastic net.Source: §7.1.10 · p. 30

8 Conclusion

The conclusion presents PINN and co-kriging approaches for degradation diagnosis without late-life aging data and compares them with delta learning and data augmentation. It reports stronger PIML performance than purely data-driven baselines, especially for late-life extrapolation, and offers method-selection guidance.

8 Conclusion

  • The study introduced PINN and co-kriging approaches for diagnosing cell degradation without relying on late-life aging data, and compared them with delta learning and data augmentation.The evaluation used data from a long-term cycling experiment.Source: §8 · p. 31
  • The PIML approaches outperformed purely data-driven baselines, with the clearest improvement in late-life degradation extrapolation.The study also conducted sensitivity analysis of PINN and co-kriging and discussed each method’s advantages and limitations.Source: §8 · p. 31
  • The study provides guidance on choosing a PIML method based on its comparative analysis and the reviewed literature.The authors frame this as a contribution to battery degradation diagnostics and PIML method selection.Source: §8 · p. 32

A1. Degradation quantification using a half-cell model

The appendix describes a half-cell model that reconstructs a pseudo-full-cell OCV curve from positive- and negative-electrode half-cell curves. It also outlines a manual fitting procedure that adjusts active masses and curve slippages to match experimental features.

A1. Degradation quantification using a half-cell model

  • The pseudo-full-cell OCV construction uses half-cell curves, cell capacity, active masses, and electrode-specific slippage parameters.The subscripts p and n denote positive and negative electrodes; mp and mn are active masses, and δp and δn quantify horizontal curve slippage.Source: A1. Degradation quantification using a… · p. 38
  • Manual fitting adjusts mn and δn to match differential-voltage peaks, then mp and δp to tune peak magnitudes and capacity-voltage endpoints.The adjustments are repeated to improve agreement between reconstructed and experimental curves.Source: A1. Degradation quantification using a… · p. 38

A2. Automatic Fitting vs. Manual Fitting

The comparison finds that PINN estimates align with manual fitting and experimental validation, while automatic fitting faces optimization and implementation challenges. The machine-learning approach reduces fitting time and automates health-parameter estimation from measured differential-voltage curves.

A2. Automatic Fitting vs. Manual Fitting

  • Automatic fitting has multiple local minima, causing run-to-run variability in active-mass estimates as the initial guess changes.The comparison summarizes the spread across five optimization runs with different initial guesses.Source: A2. Automatic Fitting vs. Manual Fitting · p. 38
  • Automatic optimization is difficult to extend with feature matching and impractical for battery-management systems when extensive iterative tuning is required.Reliable initial estimates across diverse operating conditions add further complexity.Source: A2. Automatic Fitting vs. Manual Fitting · p. 39
  • PINN active-mass estimates align with manual fitting, and both generally agree better with experimental validation than automatic fitting.The comparison used two destructively analyzed cells; the PINN was trained on manually fitted parameter estimates.Source: A2. Automatic Fitting vs. Manual Fitting · pp. 38–39

Linear kernel

The linear kernel represents the relationship between two input data points through the dot product of their feature vectors.

Linear kernel

  • The linear kernel defines similarity between two input feature vectors as their dot product.Source: Linear kernel · p. 39

Radial basis function (RBF) kernel

The section describes several kernel choices for modeling data relationships, then presents trajectory predictions and a half-cell surrogate-model training procedure.

Radial basis function (RBF) kernel

  • The RBF kernel captures smooth, nonlinear relationships by assigning greater covariance to closer points and lower covariance to farther points.Its bandwidth or length-scale parameter controls kernel width and the smoothness of the resulting function.Source: Radial basis function (RBF) kernel · p. 39

Exponential kernel

  • The exponential kernel also captures smooth, nonlinear relationships, but its faster decay suits data expected to become less correlated with distance.Figure 15 compares PINN, manual and automatic fitting, and experimental validation of active-mass estimates for two selected cells.Source: Exponential kernel · p. 39, Figure 15 · p. 40

Rational quadratic kernel

Polynomial kernel

  • The polynomial kernel generalizes the linear kernel to model polynomial relationships, with degree d and optional constant c controlling the expansion.Source: Polynomial kernel · p. 41

Matérn 3/2 kernel

  • The Matérn 3/2 kernel models moderately rough data with abrupt variations while retaining some smoothness, and is computationally efficient for moderate datasets.ρ is its length-scale parameter.Source: Matérn 3/2 kernel · p. 41

Matérn 5/2 kernel

  • The Matérn 5/2 kernel is smoother than Matérn 3/2 and can model data with greater roughness or irregularity.It has smoothness parameter ν = 5/2 and is differentiable twice; ρ is its length-scale parameter.Source: Matérn 5/2 kernel · p. 41

A4. Trajectory prediction results

A5. Half-cell surrogate model

  • Perturbed half-cell parameters and their simulated health and peak-position outputs provide training pairs for a surrogate neural network.The parameters are sampled from a 15% perturbation range; the network maps them to capacity, lithium-inventory indicator, and two dQ/dV peak voltages.Source: A5. Half-cell surrogate model · p. 46, Figure 20 · p. 46

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