Optimal Signal Decomposition-based Multi-Stage Learning for Battery Health Estimation

Bor Kiat Ng; King Jet Tseng; Qingyu Yan; Vijay Babu Pamshetti; Wei Zhang

REVIEW 4 major objections 4 minor 13 references

Optimal Signal Decomposition-based Multi-Stage Learning for Battery Health Estimation

T0 review · 4 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read PSO-tuned VMD plus CNN-LSTM estimates battery health to 0.26% mean error.

desk verdict Competent and honest engineering paper, but the headline 0.26% error is inflated by a non-causal VMD decomposition that leaks future capacity information into the input. read the letter →

arxiv 2501.16377 v1 pith:5IZ4K6YG submitted 2025-01-24 cs.LG cs.AI

Vijay Babu Pamshetti , Wei Zhang , King Jet Tseng , Bor Kiat Ng , Qingyu Yan This is my paper · ORCID

classification cs.LGcs.AI

keywords stateofhealthestimationvariationalmodedecompositionparticleswarmoptimizationCNN-LSTMbatterycapacitydegradationsignalmanagementsystem

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proposes OSL, a battery state-of-health estimator that first decomposes the capacity curve into intrinsic mode functions using variational mode decomposition whose parameters are tuned per battery by particle swarm optimization, then feeds windows of those IMFs through a two-stage CNN-LSTM. The central claim is that this combination of optimized signal decomposition and multi-stage learning yields the lowest estimation errors among all compared methods on a public four-battery aging dataset, with a best mean absolute percentage error of 0.26% on one cell and the best score in all twelve reported battery-metric cases. The authors argue this matters because practical battery management systems need accurate, low-cost health estimates, and their pipeline is simple enough, using capacity alone as input, to be deployed.

What carries the argument

The load-bearing mechanism is PSO-optimized variational mode decomposition. VMD solves a constrained variational problem to split the capacity signal into K intrinsic mode functions with compact, minimally overlapping frequency bands; its two most influential parameters, K (the number of IMFs) and alpha (the balance between band compactness and reconstruction fidelity), are tuned per battery by particle swarm optimization using Shannon entropy of the IMF envelopes as the fitness function. The resulting three IMFs become the input to a two-stage CNN-LSTM in which 1D convolutions extract spatial patterns and LSTM captures temporal dependencies.

What would settle it

Run OSL with a strictly causal VMD, refitting the decomposition online using only cycles up to t on the same four batteries and comparing MAPE; if the advantage over baselines shrinks or the 0.26% best-case error rises substantially, the non-causal preprocessing is the real source of the accuracy gain.

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Extended reading notes

Core claim

OSL's discovery is that treating battery capacity as a signal with separable frequency content, rather than as a raw time series, materially improves state-of-health estimation. With VMD decomposition optimized per battery (optimal K=3 and battery-specific alpha found by PSO using envelope entropy as the fitness), the three IMFs separate the long-term degradation trend, intermediate variations, and high-frequency noise. Feeding windows of these IMFs into a convolutional stage for spatial features and then an LSTM stage for temporal dependencies yields mean absolute errors below 0.35% across the four cells, outperforming LSTM, BiLSTM, SAGPCN, EMD-LSTM, VMD-LSTM, and VMD-Transformer in every metric-battery combination.

Load-bearing premise

The VMD decomposition is computed from the entire capacity curve before any windowing, so the IMF values fed to the network at time t contain information from future cycles; if the pipeline were constrained to causal inputs only, the reported errors could be higher.

Editorial extensions

If this is right

If OSL is right, signal decomposition plus two-stage learning is the performance frontier for capacity-only state-of-health estimation on this dataset.
VMD parameters must be optimized per battery; fixed parameters degrade results, but the choice of meta-heuristic used for tuning matters little.
Noise and rapid capacity-regeneration oscillations can be isolated into high-frequency IMFs, letting the learner focus on the degradation trend.
The same feature pipeline can be extended to future health forecasting by shifting the output window.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The paper leaves unspecified whether VMD is fit on the full capacity history before windowing; a causal implementation that recomputes IMFs with data only up to time t would likely raise the reported errors, so the headline accuracy is an upper bound under real-time constraints.
A stronger test of the method would be training on a broader set of cycling protocols and cell chemistries, since the per-battery alpha values suggest sensitivity to aging mode.
The capacity-only input is a deliberate simplification; using voltage and temperature windows as additional signal channels could either enhance the decomposition or complicate it.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The thing to know: this is a competent, honest engineering paper whose central accuracy claim is likely inflated by causal leakage. The 0.26% MAPE in the abstract is for battery B0005 only, and the input IMFs at time t are computed by VMD over the whole capacity curve, including future cycles. So as written, the experiment tests offline curve fitting, not the online estimation the paper motivates.

What is actually new is modest: PSO-optimized VMD combined with a two-stage CNN-LSTM. The authors explicitly disclaim novelty for using PSO on VMD parameters and even show that several metaheuristics converge to the same optimum. That is a genuinely useful negative result and the write-up is refreshingly clear about it. The paper is well structured, the comparisons are reasonably broad, and the claim that signal decomposition helps is supported by the consistent gap between LSTM and VMD-LSTM. The detailed PSO settings and the decomposition figures help reproducibility.

The soft spots are load-bearing, though. Equation (1) is a global variational problem, and Figure 4 shows IMFs computed across all cycles. When you feed a window ending at t-1, the IMF values inside that window already contain information from after t. The PSO optimization of alpha also uses the full test battery curve, so the test battery's decomposition parameters are tuned with future access. That is a second leakage path. The reported advantage over VMD-LSTM and VMD-Transformer could shrink or disappear under a causal decomposition with training-only parameter selection. A persistence baseline is also missing; for slowly decaying capacity, a last-value extrapolation often gives very low error. The abstract's "mean error of just 0.26%" is misleading because it is a single-battery result, not a mean over the four batteries. Minor issues: no code or data release, no confidence intervals over repeated training runs.

Who is this for? Researchers working on battery state-of-health estimation with signal decomposition. The paper is a useful cautionary example of how easy it is to leak future information in feature construction, and the metaheuristic comparison is worth having on record.

Recommendation: send it to peer review as a borderline-major revision. The topic is relevant and the authors are honest, but the evaluation protocol must be fixed before the accuracy claim is credible. I would not cite the 0.26% number until the authors rerun with causal VMD, train-only parameter tuning, a persistence baseline, and honest per-battery averages in the abstract.

Referee Report

4 major / 4 minor

Summary. The paper proposes OSL, a battery state-of-health (SoH) estimation method that combines particle-swarm-optimized variational mode decomposition (VMD) with a two-stage CNN-LSTM. The authors optimize VMD parameters K and alpha using envelope entropy, decompose the battery capacity curve into three IMFs, and feed windows of these IMFs into a CNN followed by an LSTM to estimate SoH. Experiments on the NASA battery dataset with four batteries are reported, and the paper claims that OSL achieves the best accuracy among the compared methods, including a MAPE of 0.26% on battery B0005 and best results across all twelve reported metric/battery combinations.

Significance. If the reported accuracy were achievable in a genuinely online setting, the proposed combination of optimized decomposition and multi-stage learning would be a practically useful contribution to battery health estimation. The paper also contains a useful negative result: the authors honestly observe that different meta-heuristics converge to the same VMD parameters, so the choice of optimizer is not the source of performance gains. The comparison against several baselines, including EMD-based and Transformer-based alternatives, is informative. However, the significance of the central performance claim is currently undermined by a causal-leakage issue in the evaluation protocol, so the contribution cannot yet be taken at face value.

major comments (4)

[Section II.C, Section II.D, Fig. 4] The VMD decomposition is applied to the entire battery capacity curve before any input window is formed. As stated in Section II.C and shown in Fig. 4, the three IMF values at time t are computed from the full capacity curve, including future cycles, because the variational problem in Eq. (1) is solved globally over the whole signal. The CNN-LSTM then consumes IMFs over the window t-T_in to t-1, as described in Section II.D. Therefore the input features at time t encode information from cycles after t, making the reported RMSE/MAE/MAPE an offline curve-fitting result rather than an online SoH estimation of the kind motivated in Section II.A. The authors should either provide a causal decomposition (e.g., using only data up to t) or explicitly restrict the claims to offline analysis; without this, the headline accuracy numbers do not support the claimed real-world BMS integration.
[Section III.C] The VMD parameters K and alpha are optimized per test battery using its full capacity curve. For battery B0005, for example, alpha=30 is selected by PSO running on the entire B0005 capacity signal, which includes future information and the label being predicted. In the leave-one-battery-out setup, this constitutes leakage: the test battery's decomposition parameters are not selected from training data alone. The authors should select parameters using only the training batteries or via nested cross-validation, and report whether the accuracy advantage persists under that protocol.
[Table I and Abstract] The abstract states that OSL demonstrates 'a mean error of just 0.26%', but the 0.26% MAPE is achieved only on battery B0005. The mean MAPE across the four batteries in Table I is approximately 0.33%, and the conclusion similarly refers to '0.26% MAE' while the table reports 0.26% as MAPE. The summary statistics and the metric used should be reported consistently and should not conflate a single-battery result with a dataset-wide mean.
[Section III.D.2, Table I] Results for SAGPCN, BiLSTM, and VMD-Transformer are taken from previous publications [2], [4], and [10], while the other methods are implemented by the authors. If those external results use different train/test splits, input features, or evaluation protocols than the authors' own experiments, the comparison in Table I is not apples-to-apples. The paper should state whether the same leave-one-battery-out protocol and capacity-only input were used for all compared methods, or should rerun the comparisons under a unified protocol.

minor comments (4)

[Abstract and Section I] There are several typographical errors, including 'o ptimal' in the abstract and 'B00005' instead of 'B0005' in Section III.C; these should be corrected.
[Section III.C, Fig. 3] Fig. 3 reports convergence of envelope entropy for battery B0005 only, and the text says results for other batteries are 'nearly the same'. Showing the actual convergence curves or reporting the obtained K and alpha for all batteries in a table would make the parameter-optimization claim more reproducible.
[Section II.D] The paper says the CNN input is 'the three IMFs from time t-T_in to time t-1' but does not clearly state what happens at the beginning of the series, when fewer than T_in previous IMFs exist; a description of the padding or truncation strategy would improve clarity.
[References] Reference [12] misspells 'Prognostics' as 'Progmostics', and reference [13] is listed as a 2025 arXiv paper without a DOI; the citation details should be checked.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: OSL's VMD-parameter optimization targets envelope entropy, not SoH error, and no load-bearing claim reduces to its inputs.

full rationale

The paper's central claim is empirical rather than derivational, and none of the derivation steps reduce to the reported target by construction. VMD parameters K and α are optimized using a PSO fitness function defined as envelope entropy in Eq. (3), which is computed from IMF envelopes and does not involve the SoH labels or the reported MAE/RMSE/MAPE; therefore the 'optimal' decomposition is not fitted to the evaluation metric. The input features are the IMFs of the capacity signal, while the output is SoH, and the paper states that SoH is not part of the input. The comparison baselines, including SAGPCN, BiLSTM, and VMD-Transformer, are taken from independent external papers [2], [4], and [10], and the authors' own earlier works cited as [3] and [13] are contextual and not load-bearing for the performance claims. The only substantive concern, that VMD is applied to the entire capacity curve before windowing and that per-battery α is selected using the full test-battery signal, is a data-leakage or evaluation-validity issue rather than circularity: the leaked information is future capacity values, not the target SoH values, and no equation or fitted parameter makes the reported 0.26% error equal to an input by construction. Accordingly, no circular step meeting the required evidentiary standard can be exhibited.

Assumptions & free parameters 3 free parameters · 3 assumptions · 0 invented entities

The central claim rests mainly on the choice of VMD parameters and the evaluation protocol, rather than on new theoretical entities. The key unvalidated assumption is that envelope entropy minimization leads to better SoH estimation.

free parameters (3)

VMD mode count K = 3
Chosen by PSO with envelope entropy fitness; reported as the optimal number of IMFs for all four batteries.
VMD balance parameter alpha = 30, 19, 92, 10 for B0005, B0006, B0007, B0018
Optimized per battery by PSO. Using the test battery's own alpha, computed from its full curve, may constitute information leakage in the leave-one-out evaluation.
Input window length T_in = not reported
The window length is defined in Section II.A but its value is never given, so the exact input configuration is unknown.

assumptions (3)

standard math VMD solves the constrained variational problem in Eq. (1) with the augmented Lagrangian in Eq. (2).
Taken from the VMD paper [8]; assumed correct.
domain assumption Decomposing the capacity signal into frequency bands separates noise from health-relevant trends and improves SoH estimation.
This is the core motivation; the paper argues it by showing VMD-LSTM beats LSTM, but it does not test whether the decomposition is causal or whether a simpler filter would do the same.
ad hoc to paper Minimizing envelope entropy (Eq. 3) via PSO yields VMD parameters that improve downstream SoH estimation.
The fitness function is not shown to correlate with estimation error; the paper only shows convergence of the fitness value, not a comparison with non-optimized parameters on the final metric.

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Cite this review

Pith. "Pith review of Optimal Signal Decomposition-based Multi-Stage Learning for Battery Health Estimation." pith.science (2026). https://pith.science/paper/5IZ4K6YG

@misc{pith2026250116377,
  author       = {Pith},
  title        = {Pith review of: Optimal Signal Decomposition-based Multi-Stage Learning for Battery Health Estimation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5IZ4K6YG}},
  note         = {Machine review of arXiv:2501.16377}
}

read the original abstract

Battery health estimation is fundamental to ensure battery safety and reduce cost. However, achieving accurate estimation has been challenging due to the batteries' complex nonlinear aging patterns and capacity regeneration phenomena. In this paper, we propose OSL, an optimal signal decomposition-based multi-stage machine learning for battery health estimation. OSL treats battery signals optimally. It uses optimized variational mode decomposition to extract decomposed signals capturing different frequency bands of the original battery signals. It also incorporates a multi-stage learning process to analyze both spatial and temporal battery features effectively. An experimental study is conducted with a public battery aging dataset. OSL demonstrates exceptional performance with a mean error of just 0.26%. It significantly outperforms comparison algorithms, both those without and those with suboptimal signal decomposition and analysis. OSL considers practical battery challenges and can be integrated into real-world battery management systems, offering a good impact on battery monitoring and optimization.

Figures

Figures reproduced from arXiv: 2501.16377 by the authors.

**Figure 2.** Capacity degradation curves of the four batteries in the NASA dataset. [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** The convergence of the fitness value based on envelop entropy with [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Signal decomposition with VMD in Fig. 4(a) and EMD in Fig. 4(b) [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

13 extracted references · 12 canonical work pages

[2]

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J. Zhang, J. Tian, Y . Jiang et al. , “Sagpcn: Self-attention graph pooling convolutional network for battery state of health estimation,” in 2023 IEEE 3rd International Conference on Industrial Electronics for Sus- tainable Energy Systems (IESES) , 2023, pp. 1–6

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P. Jiang, T. Zhang, G. Huang et al. , “An end-cloud collaboration approach for state-of-health estimation of lithium-ion batteries based on bi-lstm with collaboration of multi-feature and attention mechanism,” International Journal of Green Energy , vol. 21, no. 10, pp. 2205–2217, 2024

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2025 arXiv

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [2]

Sagpcn: Self-attention graph pooling convolutional network for battery state of health estimation,

J. Zhang, J. Tian, Y . Jiang et al. , “Sagpcn: Self-attention graph pooling convolutional network for battery state of health estimation,” in 2023 IEEE 3rd International Conference on Industrial Electronics for Sus- tainable Energy Systems (IESES) , 2023, pp. 1–6

work page 2023

[2] [4]

An end-cloud collaboration approach for state-of-health estimation of lithium-ion batteries based on bi-lstm with collaboration of multi-feature and attention mechanism,

P. Jiang, T. Zhang, G. Huang et al. , “An end-cloud collaboration approach for state-of-health estimation of lithium-ion batteries based on bi-lstm with collaboration of multi-feature and attention mechanism,” International Journal of Green Energy , vol. 21, no. 10, pp. 2205–2217, 2024

work page 2024

[3] [10]

Edge–cloud collaborative estimation lithium-ion battery soh based on mewoa-vmd and transformer,

Y . Chen, X. Huang, Y . He et al. , “Edge–cloud collaborative estimation lithium-ion battery soh based on mewoa-vmd and transformer,” Journal of Energy Storage , vol. 99, p. 113388, 2024

work page 2024

[4] [1]

Battery 2030: Resilient, sustainable, and circular,

J. Fleischmann, M. Hanicke, E. Horetsky et al., “Battery 2030: Resilient, sustainable, and circular,” McKinsey & Company , pp. 2–18, 2023

work page 2023

[5] [3]

Practical battery health monitoring using uncertainty-aware bayesian neural network,

Y . Zhao, W. Zhang, Q. Yan et al. , “Practical battery health monitoring using uncertainty-aware bayesian neural network,” in 2024 IEEE 100th V ehicular Technology Conference (VTC2024-Fall), 2024, pp. 1–6

work page 2024

[6] [5]

Optimized data-driven approach for remaining useful life prediction of lithium-ion batteries based on sliding window and systematic sampling,

S. Ansari, A. Ayob, M. H. Lipu et al., “Optimized data-driven approach for remaining useful life prediction of lithium-ion batteries based on sliding window and systematic sampling,” Journal of Energy Storage , vol. 73, p. 109198, 2023

work page 2023

[7] [6]

State of health estimation and re- maining useful life prediction for lithium-ion batteries by improved particle swarm optimization-back propagation neural network,

Y . Ma, M. Yao, H. Liu et al. , “State of health estimation and re- maining useful life prediction for lithium-ion batteries by improved particle swarm optimization-back propagation neural network,” Journal of Energy Storage , vol. 52, p. 104750, 2022

work page 2022

[8] [7]

A data-driven approach with uncertainty quantification for predicting future capacities and remaining useful life of lithium-ion battery,

K. Liu, Y . Shang, Q. Ouyang et al. , “A data-driven approach with uncertainty quantification for predicting future capacities and remaining useful life of lithium-ion battery,” IEEE Transactions on Industrial Electronics, vol. 68, no. 4, pp. 3170–3180, 2021

work page 2021