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arxiv: 2605.01693 · v1 · submitted 2026-05-03 · 📡 eess.SY · cs.SY

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

Khalid Mahmud Labib , Inayat Rasool , Shabbir Ahmed This is my paper

Pith reviewed 2026-05-09 17:16 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords lithium-ion batteryvoltage dynamicsdynamic mode decompositiontransformer sequence modelHPPC testdata-driven modelingbattery state estimationoperator-theoretic methods
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The pith

DMDc identifies more accurate and robust state-space models of lithium-ion battery voltage from limited HPPC data than a physics-guided transformer.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper compares an operator-theoretic approach using dynamic mode decomposition with control to a modified PatchTST transformer for predicting terminal voltage responses in a lithium-ion cell. It tests both on hybrid pulse power characterization measurements from a single 30 Ah cell and finds that delay-embedded DMDc yields lower prediction errors while remaining more stable as the cell degrades. The transformer separates open-circuit voltage analytically from a learned residual and incorporates future current inputs, achieving qualitatively similar pulse dynamics but with greater flexibility in architecture. Accurate models matter because they support battery state estimation, control, and health monitoring under varying conditions and aging. The work shows operator methods can be preferable when data is scarce, while neural sequence models may scale with larger datasets.

Core claim

In modeling nonlinear voltage dynamics of a lithium-ion battery from HPPC data, the DMDc model constructed directly from delay-embedded snapshots of terminal voltage and current produces an interpretable linear state-space representation that achieves lower prediction error and greater robustness to cell degradation than a physics-guided transformer, which decomposes voltage into an analytically computed open-circuit-voltage component plus a learned dynamic residual with tailored current fusion, at least under the limited-data regime of the experiments.

What carries the argument

DMDc operator identification from delay-embedded voltage and current snapshots, which directly yields system matrices for recursive multi-step voltage prediction without iterative training.

If this is right

  • DMDc supplies an interpretable linear model usable for real-time recursive prediction and control design in battery management systems.
  • The physics-guided transformer separation of open-circuit voltage from residuals allows modular incorporation of known electrochemical relations into sequence models.
  • Operator-theoretic methods can deliver computational efficiency and robustness advantages when training data from individual cells is limited.
  • Transformer architectures gain value for battery modeling once larger, more diverse datasets across cells and conditions become available.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • DMDc models could be embedded directly in embedded battery management hardware without retraining overhead as cells age.
  • The flexibility noted for the transformer suggests hybrid approaches that start with DMDc for baseline accuracy and add nonlinear corrections only when data volume permits.
  • Similar operator-versus-sequence comparisons could be run on other electrochemical systems such as fuel cells or supercapacitors to test whether the robustness pattern generalizes.
  • Extending the DMDc embedding to include temperature or state-of-health indicators might further improve long-term prediction without increasing model complexity.

Load-bearing premise

Data and hyperparameters from tests on one 30 Ah cell under specific HPPC profiles are sufficient to establish that DMDc is generally more accurate and robust than the transformer approach.

What would settle it

Running the same comparison on HPPC data collected from multiple cells at different stages of degradation and finding that the transformer achieves equal or lower root-mean-square prediction error than DMDc.

Figures

Figures reproduced from arXiv: 2605.01693 by Inayat Rasool, Khalid Mahmud Labib, Shabbir Ahmed.

Figure 1
Figure 1. Figure 1: Schematic of the DMDc modeling framework view at source ↗
Figure 2
Figure 2. Figure 2: Proposed physics-guided PatchTST for multi-step view at source ↗
Figure 3
Figure 3. Figure 3: Comparison of RSS for different delay embeddings (a) view at source ↗
Figure 4
Figure 4. Figure 4: Comparison of experimental measurements and mode view at source ↗
Figure 5
Figure 5. Figure 5: Change of modal magnitude with cycles Algorithm 1 Chunked autoregressive inference for physics-guided PatchTST Require: Current I1:T , measured voltage V1:T , time stamps t1:T , fitted OCV model, trained PatchTST forecaster, context length L, chunk length H, evaluation start index s Ensure: Reconstructed voltage forecast Vˆ 1:T 1: Compute pseudo-SOC trajectory SOC1:T using Eqs. (11)–(12) 2: Compute VOCV,1:… view at source ↗
read the original abstract

Lithium-ion batteries exhibit nonlinear voltage dynamics across varying operating conditions and aging states, making accurate modeling essential for estimation, control, and health monitoring. This work compares two data-driven frameworks for modeling voltage responses from hybrid pulse power characterization (HPPC) measurements: an operator-theoretic model based on Dynamic Mode Decomposition with control (DMDc), and a physics-guided transformer-based sequence model. In the DMDc framework, delay-embedded snapshots of terminal voltage and current are used to identify system matrices directly from measurement data, yielding an interpretable state-space model for recursive prediction. In parallel, a modified PatchTST architecture is developed in which terminal voltage is decomposed into an analytically computed open-circuit-voltage (OCV) component and a learned dynamic residual, with a future-current fusion pathway tailored to the prescribed HPPC current profile. Experimental results on a 30 Ah lithium-ion cell show that, although both models capture the sharp transient pulse dynamics, DMDc achieves lower prediction error and greater robustness to cell degradation under the present limited data regime, while the transformer captures qualitatively similar dynamics with greater architectural flexibility. These results highlight the advantages of operator-theoretic models in interpretability, computational efficiency, and robustness, while indicating the promise of physics-guided transformer models when larger and more diverse datasets are available.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript compares two data-driven frameworks for modeling lithium-ion battery voltage dynamics from HPPC measurements on a 30 Ah cell: an operator-theoretic DMDc model that uses delay-embedded snapshots of terminal voltage and current to identify linear state-space matrices for recursive prediction, and a physics-guided PatchTST transformer that analytically subtracts OCV and learns the dynamic residual with a future-current fusion pathway. Experiments indicate that both capture sharp transient pulse dynamics, but DMDc achieves lower prediction error and greater robustness to cell degradation in the limited-data regime, while the transformer offers greater architectural flexibility.

Significance. If the comparative results hold under broader validation, the work provides concrete evidence for the advantages of DMDc in interpretability, computational efficiency, and robustness for battery voltage modeling under data scarcity, while highlighting the promise of physics-guided sequence models when larger datasets become available. This contributes to systems and control applications in battery management by directly contrasting operator-theoretic and deep-learning approaches on real cell data.

major comments (2)
  1. [Experimental Results] Experimental validation section: The central claims that DMDc achieves lower prediction error and greater robustness to degradation (relative to the physics-guided PatchTST) rest on results from a single 30 Ah cell's HPPC dataset. No cross-validation across multiple cells, no reported variance across random seeds or data splits, and no error bars or statistical tests are mentioned, which leaves open the possibility that the observed edge is dataset-specific rather than intrinsic (e.g., due to alignment with DMDc's linear structure or chosen delay-embedding length and DMD rank).
  2. [Methods / Experimental Results] Model training and evaluation: Both the DMDc identification (fitting system matrices to delay-embedded snapshots) and the transformer residual learning (on OCV-subtracted measurements with current fusion) use the same limited HPPC pulses for training and performance assessment. The manuscript should clarify whether an independent validation set or out-of-sample aging trajectories were employed, as post-fit metrics on the training distribution do not fully harden the robustness and superiority claims.
minor comments (2)
  1. [Abstract / Results] The abstract and results text should explicitly state how cell degradation is quantified (e.g., capacity fade percentage or cycle count) and at which specific aging points the models were evaluated to support the robustness comparison.
  2. [Methods] Notation for the DMDc state-space matrices and the transformer's embedding/patch parameters could be clarified with a table summarizing the free hyperparameters (delay length, DMD rank, PatchTST hyperparameters) and how they were selected.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on our manuscript. We address each major comment below with clarifications and revisions to strengthen the experimental presentation.

read point-by-point responses

  1. Referee: Experimental validation section: The central claims that DMDc achieves lower prediction error and greater robustness to degradation (relative to the physics-guided PatchTST) rest on results from a single 30 Ah cell's HPPC dataset. No cross-validation across multiple cells, no reported variance across random seeds or data splits, and no error bars or statistical tests are mentioned, which leaves open the possibility that the observed edge is dataset-specific rather than intrinsic (e.g., due to alignment with DMDc's linear structure or chosen delay-embedding length and DMD rank).

    Authors: We acknowledge the limitation of using data from a single 30 Ah cell, which restricts broader generalizability. The robustness to degradation is evaluated using HPPC measurements collected at multiple aging stages within this cell's dataset, serving as a proxy for out-of-sample aging trajectories. To improve statistical rigor, we will add error bars for the transformer by reporting mean and standard deviation over multiple random seeds, and include sensitivity analysis for DMDc rank selection. The Experimental Results section has been revised to incorporate these elements and explicitly discuss the single-cell scope. revision: partial

  2. Referee: Model training and evaluation: Both the DMDc identification (fitting system matrices to delay-embedded snapshots) and the transformer residual learning (on OCV-subtracted measurements with current fusion) use the same limited HPPC pulses for training and performance assessment. The manuscript should clarify whether an independent validation set or out-of-sample aging trajectories were employed, as post-fit metrics on the training distribution do not fully harden the robustness and superiority claims.

    Authors: We have revised the manuscript to clarify the data partitioning: the HPPC pulses were divided into training and held-out test sets, with test pulses excluded from DMDc matrix identification and transformer training. For degradation robustness, later-cycle aging trajectories were used as out-of-sample evaluation data. The Methods and Experimental Results sections now explicitly detail this splitting procedure and confirm independent test sets for all reported metrics. revision: yes

Circularity Check

2 steps flagged

DMDc matrix fitting and transformer residual learning both operate on the identical single-cell HPPC snapshots used to report prediction errors.

specific steps
  1. fitted input called prediction [Abstract]
    "In the DMDc framework, delay-embedded snapshots of terminal voltage and current are used to identify system matrices directly from measurement data, yielding an interpretable state-space model for recursive prediction. [...] Experimental results on a 30 Ah lithium-ion cell show that, although both models capture the sharp transient pulse dynamics, DMDc achieves lower prediction error and greater robustness to cell degradation under the present limited data regime"

    System matrices are constructed by least-squares fitting to the exact delay-embedded snapshots; the subsequent 'recursive prediction' and error metric are therefore evaluations of the fitted operator on the training measurements themselves, rendering the superiority claim a statement about relative fit quality on this single dataset rather than generalization.

  2. fitted input called prediction [Abstract]
    "a modified PatchTST architecture is developed in which terminal voltage is decomposed into an analytically computed open-circuit-voltage (OCV) component and a learned dynamic residual, with a future-current fusion pathway tailored to the prescribed HPPC current profile"

    The dynamic residual is learned by gradient descent on the identical HPPC voltage measurements (after external OCV subtraction); reported qualitative similarity and flexibility therefore reflect in-sample learning performance on the same traces used for DMDc comparison.

full rationale

The paper's central empirical claim (DMDc lower error and greater robustness) rests on post-identification metrics computed from models whose parameters are obtained by direct fitting to the same limited HPPC voltage/current traces. While OCV subtraction is external and the architectures differ, no independent test trajectory, multi-cell hold-out, or cross-validation is described in the provided text; therefore the reported superiority reduces to a comparison of in-sample reconstruction quality rather than out-of-distribution prediction. This matches the fitted-input-called-prediction pattern but does not rise to full self-definitional equivalence or load-bearing self-citation, yielding a moderate circularity score.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

Claims rest on standard DMDc linear identification and transformer training assumptions plus OCV subtraction; no new physical entities postulated.

free parameters (2)
  • delay embedding length and DMD rank
    Chosen to capture dynamics from HPPC snapshots; affects identified matrices.
  • PatchTST hyperparameters and training schedule
    Control learned residual; tuned on the limited dataset.
axioms (2)
  • domain assumption Voltage dynamics approximable by finite-dimensional linear operator on delay-embedded snapshots.
    Invoked for DMDc data matrix construction.
  • domain assumption Open-circuit voltage known analytically and subtractable to isolate dynamic residual.
    Used in transformer to inject physics.

pith-pipeline@v0.9.0 · 8338 in / 1244 out tokens · 38109 ms · 2026-05-09T17:16:37.964582+00:00 · methodology

discussion (0)

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