PiDDM：物理信息可微退化建模用于锂离子电池健康状态预测

A purely data-driven approach could not guarantee these properties. Therefore, the final degradation-rate term is expressed as a separable function of operating conditions and time/throughput, multiplied by an Arrhenius acceleration factor. This formulation is differentiable with respect to all relevant variables and can be incorporated into the loss function as a soft constraint that biases the neural network toward physically plausible degradation trajectories.

### 2.2 Model architecture and training

The overall architecture of PiDDM is illustrated in Figure 1. The model takes as input a set of operating-condition features, including cycling protocol identifiers, temperature, depth of discharge, C-rate, and accumulated charge throughput. A multi-layer perceptron (MLP) with several hidden layers maps these inputs to a predicted SOH value. The degradation-regularization term is added to the standard mean-squared-error loss. During training, the network learns to minimize both the data-fitting error and the deviation from the physics-informed degradation trend. This allows PiDDM to exploit the inductive bias provided by the Arrhenius-type degradation kinetics while retaining the flexibility of a neural network to adapt to specific battery behaviors.

**Figure 1:** Schematic of the PiDDM framework. Operating-condition features are fed into an MLP, and the output is regularized by a physics-informed degradation term based on SEI-growth kinetics.

The training procedure is as follows. The dataset is split into training and test sets at the battery level, meaning that all cycles from a given battery are either in the training set or the test set. This protocol-level split evaluates the model's ability to generalize across different operating conditions. All neural networks are trained using the Adam optimizer for a fixed number of epochs with a learning rate schedule. The loss function for PiDDM is defined as

\[
\mathcal{L} = \mathcal{L}_{\text{data}} + \lambda \mathcal{L}_{\text{phys}},
\]

where \(\mathcal{L}_{\text{data}}\) is the mean-squared-error between predictions and ground-truth SOH values, \(\mathcal{L}_{\text{phys}}\) is the physics-informed degradation penalty, and \(\lambda\) is a weighting factor that balances the two terms.

The physics-informed penalty is computed by evaluating the gradient of the predicted SOH with respect to time. The penalty encourages this gradient to be negative (capacity fade) and to have a magnitude that follows the Arrhenius-type degradation rate. This is implemented by penalizing instances where the predicted SOH increases, or where the rate of decrease deviates significantly from the physics-based expectation. By doing so, PiDDM explicitly prevents the non-physical capacity regeneration observed in unregularized models.

## 3 Results and discussion

### 3.1 Dataset description and evaluation protocol

The proposed model is evaluated on a public dataset containing 55 commercial lithium-ion batteries cycled under six distinct operating protocols. The protocols include constant-current charging and discharging, constant-current charging with dynamic discharging, and more complex load profiles that simulate realistic usage patterns. Batteries in the dataset were cycled until end-of-life, defined as 80% of initial capacity. The dataset provides cycle-level capacity measurements, which are used to compute SOH as the ratio of current capacity to initial capacity.

For the main prediction task, models are trained to predict SOH from operating-condition features. The training set contains batteries from all six protocols, and the test set contains held-out batteries from the same protocols. For the extrapolation task, models are trained on the first 90% of each battery's cycle life and evaluated on the remaining 10%, which typically corresponds to the accelerated degradation phase near end-of-life.

### 3.2 Overall prediction accuracy

Table 1 summarizes the average mean-squared-error (MSE) and mean-absolute-error (MAE) for the different models. PiDDM achieves the lowest average prediction error among all methods. Compared with the standard MLP, PiDDM reduces the MSE by a substantial margin, indicating that the physics-informed regularization provides useful inductive bias beyond what can be learned from finite training data. PiDDM also outperforms the baseline PINN, demonstrating that embedding domain-specific degradation kinetics is more effective than using a generic physical residual penalty.

**Table 1:** Comparison of average prediction errors across models.

| Model | MSE × 10⁻⁴ | MAE × 10⁻² |
|-------|------------|------------|
| MLP   | 3.21       | 1.24       |
| PINN  | 2.85       | 1.16       |
| PiDDM | 2.47       | 1.08       |

The improvement of PiDDM over the MLP and PINN baselines is consistent across individual cycling protocols, with the largest gains observed for protocols involving dynamic load profiles where the relationship between operating conditions and degradation is more complex. This result suggests that the physics-informed degradation term helps the model identify the most relevant features and generalize more effectively.

### 3.3 Extrapolation performance near end-of-life

Figure 2 shows representative SOH predictions for the extrapolation task, where models are trained on the first 90% of cycle life and evaluated on the remaining 10%. In this setting, the ground-truth SOH typically exhibits an accelerated decline toward the end of life. The MLP and PINN baselines often produce predictions that flatten or even increase during this phase, corresponding to non-physical capacity regeneration. In contrast, PiDDM captures the accelerated degradation trend more accurately. The predicted SOH continues to decrease monotonically, in agreement with the physics-informed constraint, and the end-of-life point is predicted with notably lower error than the baselines.

**Figure 2:** Extrapolation results for a representative battery. Shaded region indicates the unseen 10% of cycle life. PiDDM (red) tracks the ground-truth degradation curve more closely than the MLP and PINN baselines, particularly in the accelerated end-of-life region.

Quantitatively, PiDDM reduces the average extrapolation MSE by over 30% compared with the MLP baseline. The elimination of capacity regeneration in the predictions is particularly notable: while the baseline models exhibit non-physical SOH increases in a significant fraction of test batteries, PiDDM does not produce such behavior in any of the tested cases. These results demonstrate that the physics-informed degradation regularization provides both quantitative and qualitative improvements in prediction quality.

### 3.4 Sensitivity to the regularization weight

The weighting factor \(\lambda\) in the loss function controls the strength of the physics-informed penalty. We investigate the sensitivity of PiDDM to this hyperparameter by varying \(\lambda\) over a range of values. As shown in Figure 3, the prediction error initially decreases as \(\lambda\) increases from zero, reaching a minimum at an intermediate value, and then increases as the penalty becomes too strong and over-constrains the network. Across the tested range, PiDDM maintains lower error than the MLP baseline, indicating that the method is robust to the choice of \(\lambda\) as long as it is within a reasonable range.

**Figure 3:** Sensitivity of PiDDM's prediction error to the physics-informed regularization weight \(\lambda\).

The optimal value of \(\lambda\) depends on the noise level in the data and the relative scale of the data-fitting and physics terms. In practice, \(\lambda\) can be selected using a validation set. The fact that a broad range of \(\lambda\) values leads to improved performance over the unregularized baseline suggests that the physics-informed prior is well aligned with the underlying degradation dynamics.

### 3.5 Physical consistency of predictions

Beyond quantitative error metrics, we evaluate the physical consistency of the model predictions. Specifically, we measure the fraction of test batteries for which the predicted SOH is monotonic non-increasing across all cycles. The MLP and PINN baselines produce non-monotonic predictions in a substantial fraction of test batteries, with predicted SOH increasing between consecutive cycles despite the known irreversibility of capacity fade. In contrast, PiDDM produces monotonic predictions in all test cases. This property is important for practical applications, as capacity regeneration in a health-monitoring system would undermine trust in the model and could lead to incorrect maintenance decisions.

The physical consistency of PiDDM is also reflected in the behavior of the predicted degradation rate. In the baseline models, the predicted degradation rate often fluctuates in ways that do not correlate with operating conditions in a physically meaningful way. PiDDM, by construction, produces degradation rates that follow the expected Arrhenius-type temperature dependence and increase with accumulated charge throughput. This alignment with known degradation physics makes the model more interpretable and provides a plausible causal link between operating conditions and predicted health.

## 4 Conclusion

We developed a physics-informed differentiable degradation modeling framework, PiDDM, for lithium-ion battery state-of-health prediction. The key contribution of this work is the incorporation of Arrhenius-type degradation kinetics associated with SEI growth and loss of lithium inventory into the neural network training objective. Unlike generic physics-informed neural networks that impose residual penalties on governing equations, PiDDM embeds degradation physics as a directional constraint on the temporal evolution of SOH. This design encourages monotonic capacity fade and improves predictive robustness across diverse operating protocols.

We evaluated PiDDM on a public dataset of 55 batteries cycled under six distinct protocols. In both in-distribution prediction and extrapolation tasks, PiDDM outperformed standard MLP and PINN baselines, achieving lower mean-squared-error and avoiding non-physical capacity regeneration in the predicted SOH trajectories. The model's predictions are monotonic across all test batteries, a property that is crucial for practical deployment in battery health monitoring systems. These results demonstrate that integrating degradation physics into deep learning models offers a promising path toward more accurate, robust, and physically credible battery state-of-health prediction.

Future work could extend this framework to incorporate additional degradation modes, such as active material loss and impedance rise, and to consider more detailed chemistry-specific degradation models. The framework could also be extended to predict other battery health indicators such as remaining useful life and power fade. Given its differentiability and modular design, PiDDM can be readily adapted to other battery chemistries and operating conditions, providing a general approach for physics-informed health modeling in energy storage systems.

## Acknowledgments

This work was supported in part by the U.S. Department of Energy and the National Science Foundation. The authors thank the developers of the public battery dataset for making their data available for research purposes.

## Data availability

The dataset used in this study is publicly available. The code implementing PiDDM is available from the corresponding author upon reasonable request.

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