Operator-Theoretic and physics-guided Sequence Modeling of Lithium-Ion Battery Voltage Dynamics

Technical Summary: Operator-Theoretic and Physics-Guided Sequence Modeling of Lithium-Ion Battery Voltage Dynamics

Problem Statement
Accurate modeling of lithium-ion battery voltage dynamics is critical for battery management systems (BMS) to ensure safe operation, state estimation, and health monitoring. Batteries exhibit complex nonlinear behaviors that vary with operating conditions and aging states. While physics-based models (e.g., P2D) offer high fidelity, they incur high computational costs. Equivalent circuit models (ECMs) are computationally efficient but often struggle to represent degradation-dependent dynamics without extensive parameter identification. Conversely, purely data-driven methods (e.g., SINDy, NARX) can be sensitive to noise and lack explicit state-space structures suitable for control. This study addresses the challenge of modeling voltage responses under Hybrid Pulse Power Characterization (HPPC) excitation, specifically comparing an operator-theoretic approach against a physics-guided deep learning architecture to determine their efficacy in terms of accuracy, robustness to degradation, and interpretability.

Methodology
The study evaluates two distinct data-driven frameworks using HPPC data from a 30 Ah lithium-ion cell:

1. Operator-Theoretic DMDc Model:

   • Framework: Dynamic Mode Decomposition with control (DMDc) is employed to identify linear system matrices directly from measurement data.
   • Data Representation: Delay-embedded snapshots of terminal voltage and current are constructed using Hankel matrices to lift temporal dependencies into a Koopman-observable space.
   • Formulation: The method solves for linear operators $A$ (state transition) and $B$ (input map) that satisfy the relationship $x_{k+1} \approx Ax_k + Bu_k$, where $x$ represents the embedded voltage state and $u$ represents the current input.
   • Training/Testing: The model is identified using data from the healthy state (up to cycle 100) and subsequently applied to predict voltage trajectories in degraded states (cycles 120, 140, and 160) without retraining, to assess robustness.

2. Physics-Guided Transformer-Based Sequence Model:

   • Architecture: A modified PatchTST (Patch Time Series Transformer) architecture is developed.
   • Physics-Guided Decomposition: Instead of predicting raw terminal voltage, the model predicts the dynamic residual $V_{dyn}(t)$. The total voltage is decomposed as $V(t) = V_{OCV}(SOC(t)) + V_{dyn}(t)$, where the Open-Circuit Voltage (OCV) component is analytically computed using a pseudo-State-of-Charge (SOC) derived from Coulomb counting and an empirical lookup table. This removes low-frequency thermodynamic drift, allowing the model to focus on fast transient dynamics.
   • Input Fusion: The model incorporates a "future-current fusion pathway," explicitly utilizing the known future current profile prescribed by the HPPC protocol during inference.
   • Training/Testing: The model is trained on a file-level partition of the dataset (cycles 1–100) and evaluated on held-out test files (cycles 101–160) using a chunked autoregressive inference strategy.

Key Contributions

• Comparative Analysis: The work provides a direct comparison between an interpretable, linear operator-theoretic model (DMDc) and a flexible, non-linear deep learning model (PatchTST) for battery voltage forecasting under identical HPPC conditions.
• Physics-Guided Deep Learning: It introduces a specific decomposition strategy for transformers in battery applications, separating the analytically solvable OCV component from the learned dynamic residual, and integrating known future inputs (current) into the attention mechanism.
• Degradation Robustness Assessment: The study evaluates model performance on degraded cells using operators/weights trained solely on healthy-state data, highlighting the generalization capabilities of both frameworks.
• Modal Analysis: The DMDc approach is shown to yield eigenvalues that reflect system dissipative behavior, offering a potential metric for characterizing aging.

Results

• Prediction Accuracy: Under the current limited data regime, the DMDc model outperformed the transformer-based model. For the 120th, 140th, and 160th cycles, DMDc achieved Root Mean Square Errors (RMSE) of approximately 0.011 V, whereas the transformer model achieved RMSEs around 0.021 V.
• Robustness to Degradation: The DMDc model, trained on healthy data, maintained high accuracy and captured sharp transient pulse dynamics and slow relaxation behaviors in degraded cycles with minimal deviation. The transformer model, while capturing qualitatively similar dynamics, exhibited higher error accumulation during autoregressive rollout.
• Modal Behavior: The identified DMDc eigenvalues showed a systematic decrease in modal magnitudes ($r = |\lambda|$) as the battery aged, indicating increasingly dissipative dynamics.
• Data Efficiency: The DMDc model demonstrated superior performance with fewer parameters and less diverse training data, attributed to its strong structural prior (linear controlled evolution). The transformer model, with ~3.35 million parameters, struggled to generalize robustly given the limited diversity of the HPPC dataset (single cell, single protocol).

Significance and Claims
The paper concludes that for battery voltage forecasting under HPPC excitation, the operator-theoretic DMDc approach offers a favorable balance of accuracy, robustness, interpretability, and computational efficiency. The study highlights that while physics-guided transformer models possess architectural flexibility and the potential to capture complex nonlinear dependencies, their performance in this specific context was constrained by the size and diversity of the available dataset.

The authors modestly claim that their results do not prove transformer models are inherently inferior for battery applications. Instead, they suggest that the advantages of transformer-based approaches may only be realized when trained on substantially larger, more heterogeneous datasets spanning multiple cells, aging trajectories, and operating conditions. Conversely, the study positions operator-theoretic methods as highly promising for adaptive battery diagnostics and prognostics, particularly in scenarios where data is limited or interpretability is paramount.