STEK: Structural-Terrain-Enhanced Koopman-Inspired Cross-Channel Regularization for Retrospective Battery Multi-Target Estimation

Jing Wang; Shiyuan Piao; Fugee Tsung

cs.AINov 4, 2026

STEK: Structural-Terrain-Enhanced Koopman-Inspired Cross-Channel Regularization for Retrospective Battery Multi-Target Estimation

Authors: Jing Wang, Shiyuan Piao, Fugee Tsung

Organizations: University of Science and Technology of China, Hefei, China · The Hong Kong University of Science and Technology, Hong Kong, China

Abstract

Retrospective battery characterization across heterogeneous cells is challenging because cycling records vary substantially with chemistry, protocol, cell format, capacity, and operating conditions. An endpoint charge--discharge curve captures the local condition of an evaluated cycle, while a cell-level lifecycle terrain summarizes structural evolution across the archived trajectory. We propose STEK, a structural-terrain-enhanced framework for retrospective battery multi-target estimation. STEK combines a cell-level lifecycle (dQ/dV) terrain with an endpoint-derived structural proxy through quality-proxy-conditioned gating. Bounded FiLM and a validity-gated residual head refine the endpoint-curve representation while preserving a curve-based prediction pathway. Battery-type and optional material-text representations further condition an identity-centered low-rank linear map, providing a Koopman-inspired cross-channel consistency objective without modeling temporal dynamics or performing rollout. Under the fixed seven-output MIX-battery protocol, the full retrospective setting improves aggregate performance over endpoint-only learned baselines across multiple temporal backbones; this comparison reflects both the additional archived context and the modeling design. Under matched auxiliary inputs, STEK outperforms direct concatenation and quality-unaware FiLM, while descriptor-removal tests show graceful fallback to the endpoint pathway. Deterministic rules remain preferable for directly recoverable capacity summaries. The method targets retrospective archive harmonization rather than online estimation from cycle t alone.

Figures & tables

Figure 1: Endpoint and lifecycle inputs used by STEK.

Figure 2: STEK architecture. Lifecycle context refines endpoint predictions, while structural and material context condition a training-only Koopman-inspired cross-channel regularizer.

Task	Metric	STEK	MambaLithium	LiPM	CPTransformer	iTransformer	Koopa	KNF
NDC	MAE $\downarrow$	0.90 $\pm$ 0.06	1.55 $\pm$ 0.32	2.07 $\pm$ 0.22	1.26 $\pm$ 0.03	1.16 $\pm$ 0.07	1.25 $\pm$ 0.12	1.35 $\pm$ 0.06
NDC	MaxAE $\downarrow$	7.65 $\pm$ 1.15	13.27 $\pm$ 0.74	16.16 $\pm$ 8.26	13.62 $\pm$ 1.42	12.04 $\pm$ 1.62	10.85 $\pm$ 3.98	9.34 $\pm$ 0.49
SOH	MAE $\downarrow$	0.40 $\pm$ 0.03	0.56 $\pm$ 0.02	0.65 $\pm$ 0.03	0.60 $\pm$ 0.06	0.55 $\pm$ 0.02	0.62 $\pm$ 0.05	0.63 $\pm$ 0.02
SOH	RMSE $\downarrow$	0.99 $\pm$ 0.02	1.51 $\pm$ 0.02	1.21 $\pm$ 0.03	1.50 $\pm$ 0.12	1.46 $\pm$ 0.07	1.44 $\pm$ 0.14	1.33 $\pm$ 0.04
CF	MAE $\downarrow$	0.38 $\pm$ 0.02	0.57 $\pm$ 0.02	0.65 $\pm$ 0.06	0.60 $\pm$ 0.06	0.55 $\pm$ 0.03	0.62 $\pm$ 0.04	0.64 $\pm$ 0.03
CF	RMSE $\downarrow$	0.97 $\pm$ 0.02	1.51 $\pm$ 0.01	1.20 $\pm$ 0.03	1.50 $\pm$ 0.12	1.47 $\pm$ 0.08	1.44 $\pm$ 0.13	1.34 $\pm$ 0.03

Table 1: Performance under the fixed seven-output MIX-battery protocol. STEK uses endpoint curves and archived auxiliary context; the other learned models are endpoint-only. Thus, the table compares complete settings rather than matched-input architectures; Table 3 provides matched-input controls. NDC, SOH, CF, NCC, and Overall are multiplied by

10^{2}

, and CFR by

10^{4}

. Deterministic NDC/NCC rules are reported in the supplementary material. Lower values are better.

Figure 3: Generality of STEK across temporal backbones. (a) Overall scores before and after adding STEK. (b) Relative improvement on each output’s primary metric.

		Full STEK	Reduced Modality Sets			Module Ablations
Task	Metric	Full STEK	Curve+Material + $dQ/dV$	Curve +Material	Curve +struct. Proxy	w/o FiLM	w/o Struct. Head	w/o Koopman Reg.
NDC	MAE $\downarrow$	0.90 $\pm$ 0.06	0.94 $\pm$ 0.10	1.00 $\pm$ 0.17	1.01 $\pm$ 0.02	1.06 $\pm$ 0.03	0.94 $\pm$ 0.10	0.94 $\pm$ 0.07
NDC	MaxAE $\downarrow$	7.65 $\pm$ 1.15	8.83 $\pm$ 1.14	9.27 $\pm$ 2.03	7.71 $\pm$ 2.78	8.01 $\pm$ 0.25	8.02 $\pm$ 1.29	7.97 $\pm$ 2.11
SOH	MAE $\downarrow$	0.40 $\pm$ 0.03	0.40 $\pm$ 0.03	0.40 $\pm$ 0.01	0.39 $\pm$ 0.01	0.44 $\pm$ 0.04	0.44 $\pm$ 0.04	0.40 $\pm$ 0.04
SOH	RMSE $\downarrow$	0.99 $\pm$ 0.02	0.96 $\pm$ 0.02	1.06 $\pm$ 0.12	0.96 $\pm$ 0.06	1.06 $\pm$ 0.06	1.04 $\pm$ 0.09	1.03 $\pm$ 0.13
CF	MAE $\downarrow$	0.38 $\pm$ 0.02	0.39 $\pm$ 0.02	0.40 $\pm$ 0.01	0.39 $\pm$ 0.02	0.42 $\pm$ 0.03	0.42 $\pm$ 0.04	0.40 $\pm$ 0.04

Table 2: Modality and component ablations on MIX-battery. Struct. Proxy denotes the endpoint-derived structural descriptor.

dQ/dV

denotes the cell-level lifecycle terrain. NDC, SOH, CF, NCC, and Overall are scaled by

10^{2}

, and CFR by

10^{4}

. Lower is better. Best results are shaded and bold; second-best results are bold. Highlighting is determined using the unrounded mean values.

		Full STEK	Fusion Controls		Koopman Controls
Task	Metric	Full STEK	Direct Concat	FiLM w/o Quality	Shared K.	State-Cond. K.	w/o K. Reg.
NDC	MAE $\downarrow$	0.90 $\pm$ 0.06	1.17 $\pm$ 0.21	0.98 $\pm$ 0.12	0.94 $\pm$ 0.06	0.94 $\pm$ 0.09	0.94 $\pm$ 0.07
NDC	MaxAE $\downarrow$	7.65 $\pm$ 1.15	10.02 $\pm$ 0.54	7.80 $\pm$ 0.56	8.96 $\pm$ 2.43	7.41 $\pm$ 0.44	7.97 $\pm$ 2.11
SOH	MAE $\downarrow$	0.40 $\pm$ 0.03	0.47 $\pm$ 0.04	0.42 $\pm$ 0.05	0.38 $\pm$ 0.02	0.40 $\pm$ 0.03	0.40 $\pm$ 0.04
SOH	RMSE $\downarrow$	0.99 $\pm$ 0.02	1.30 $\pm$ 0.16	0.99 $\pm$ 0.01	0.99 $\pm$ 0.03	0.97 $\pm$ 0.06	1.03 $\pm$ 0.13
CF	MAE $\downarrow$	0.38 $\pm$ 0.02	0.48 $\pm$ 0.04	0.42 $\pm$ 0.03	0.38 $\pm$ 0.02	0.40 $\pm$ 0.02	0.40 $\pm$ 0.04

Table 3: Fusion and Koopman-inspired controls on MIX-battery. Fusion controls use the same auxiliary inputs as Full STEK. Shared K. uses a shared identity-centered lifted map, whereas State-Cond. K. conditions it on the endpoint representation. Lower values are better for all metrics. NDC, SOH, CF, NCC, and Overall are scaled by

10^{2}

, and CFR by

10^{4}

. Best results are shaded and bold; second-best results are bold. Highlighting is determined using the unrounded mean values.

Figure 4: Performance–efficiency comparison under the fixed MIX-battery seven-output setting. Bubble size represents FLOPs, and lower Overall scores indicate better performance.

		CALCE		HNEI		MATR
Task	Metric	Curve	Full	Curve	Full	Curve	Full
NDC	MAE $\downarrow$	8.93 $\pm$ 0.74	8.39 $\pm$ 0.08	2.93 $\pm$ 2.17	1.90 $\pm$ 0.43	29.17 $\pm$ 8.95	13.55 $\pm$ 4.64
NDC	MaxAE $\downarrow$	14.42 $\pm$ 1.06	14.19 $\pm$ 0.61	7.25 $\pm$ 0.51	8.66 $\pm$ 3.66	43.43 $\pm$ 2.46	35.67 $\pm$ 2.47
SOH	MAE $\downarrow$	2.36 $\pm$ 0.77	1.86 $\pm$ 0.46	1.59 $\pm$ 0.06	1.60 $\pm$ 0.37	4.23 $\pm$ 0.62	2.75 $\pm$ 0.14
SOH	RMSE $\downarrow$	3.07 $\pm$ 0.83	2.74 $\pm$ 0.66	1.72 $\pm$ 0.08	1.78 $\pm$ 0.30	4.53 $\pm$ 0.50	3.29 $\pm$ 0.18
CF	MAE $\downarrow$	2.31 $\pm$ 0.76	1.86 $\pm$ 0.39	1.28 $\pm$ 0.09	1.23 $\pm$ 0.39	4.27 $\pm$ 0.67	2.44 $\pm$ 0.35

Table 4: Holdout evaluation results for TimesNet under the curve-only and full retrospective STEK settings. NDC, SOH, CF, NCC, and Overall are scaled by

10^{2}

, while CFR metrics are scaled by

10^{4}

. Results are reported as mean

\pm

standard deviation over three random seeds. Lower values are better, and the better result under each dataset is highlighted in bold.

Descriptor	MAE $\downarrow$						RQE (%) $\downarrow$	Overall $\downarrow$
missing rate	NDC	SOH	CF	CFR	ICR ( $\mathrm{m}\Omega$ )	NCC	QTY	Score
0%	0.90 $\pm$ 0.06	0.40 $\pm$ 0.03	0.38 $\pm$ 0.02	1.33 $\pm$ 0.00	6.54 $\pm$ 0.98	0.85 $\pm$ 0.03	6.74 $\pm$ 0.64	12.42 $\pm$ 0.30
25%	0.91 $\pm$ 0.05	0.40 $\pm$ 0.03	0.39 $\pm$ 0.01	1.33 $\pm$ 0.00	6.86 $\pm$ 0.91	0.87 $\pm$ 0.03	8.14 $\pm$ 0.90	12.52 $\pm$ 0.28
50%	0.92 $\pm$ 0.06	0.40 $\pm$ 0.03	0.38 $\pm$ 0.01	1.33 $\pm$ 0.00	7.09 $\pm$ 0.92	0.88 $\pm$ 0.05	9.15 $\pm$ 1.20	12.57 $\pm$ 0.28
75%	0.91 $\pm$ 0.09	0.40 $\pm$ 0.03	0.39 $\pm$ 0.02	1.33 $\pm$ 0.01	7.37 $\pm$ 0.97	0.88 $\pm$ 0.07	10.76 $\pm$ 1.73	12.65 $\pm$ 0.31
100%	0.90 $\pm$ 0.12	0.40 $\pm$ 0.03	0.39 $\pm$ 0.02	1.33 $\pm$ 0.01	7.63 $\pm$ 1.11	0.89 $\pm$ 0.10	12.15 $\pm$ 2.13	12.71 $\pm$ 0.36

Table 5: Robustness to cell-level masking of the lifecycle terrain and structural proxy. Intermediate rates use three model seeds and ten masking seeds; 0% and 100% use three deterministic runs. NDC, SOH, CF, NCC, and Overall are scaled by

10^{2}

, and CFR by

10^{4}

; ICR and QTY are reported in m

\Omega

and %.

Appendix figures & tables11 assets

Supplementary material from the paper’s appendix.

Appendix

Source	Type	Format	Cathode	Anode	Cap. (Ah)	Temp. ( ∘ C)	Cells	Cycles	Prot.
CALCE	Li-ion	Prismatic	LCO	Graphite	1.10	25	13	14,299	2
HNEI	Li-ion	18650	LCO+NCM442	Graphite	2.80	25	14	15,164	2
MATR	Li-ion	18650	LFP	Graphite	1.10	30	169	140,001	81
RWTH	Li-ion	18650	NMC	Carbon	3.00	25	48	105,006	1
MICH	Li-ion	Pouch	NCM111	Graphite	2.36	25, 45	40	19,895	2
MICH_EXP	Li-ion	Pouch	NCM111	Graphite	5.00	$-5$ , 25, 45	18	6,545	6

Appendix

Table 1: Source statistics for MIX-battery. Capacity denotes the source-reported nominal capacity, which is retained as a dataset descriptor; harmonized values used for curve normalization are specified in the preprocessing section. AG denotes artificial graphite, and Prot. denotes the number of cycling protocols.

Model	Input	Width	Principal adaptation
TimesNet	curves	64	seven-output prediction head
DLinear	curves	64	seven-output prediction head
PatchTST	curves	64	seven-output prediction head
iTransformer	curves	64	seven-output prediction head
MambaLithium	curves	64	canonical-curve adaptation
LiPM	curves	64	canonical-curve adaptation

Appendix

Table 2: Principal baseline settings used in the common seven-output comparison. Native architectural components are retained where compatible, while all models use the same prediction interface and evaluation protocol.

Method	NDC MAE	NDC MaxAE	NCC MAE	NCC RMSE
STEK–TimesNet	0.9022	7.6520	0.8507	1.3079
Segment-maximum rule	1.0121	102.2336	0.0392	0.1384
Segment maximum + zero-segment fallback	0.0276	0.8163	0.0392	0.1384

Appendix

Table 3: Input-derived capacity-recovery diagnostic on the MIX-battery test split. All errors are multiplied by

10^{2}

. The deterministic rules require neither training nor target-label access. The plain NDC rule is affected by four endpoints with an identically zero resampled discharge segment; over the remaining 410 endpoints, its NDC MAE and MaxAE are 0.0247 and 0.8163, respectively. The STEK values are included for scale, with seed-level uncertainty reported in the main comparison. Rule-based rows apply only to NDC and NCC and therefore do not receive an Overall score.

Category	Outputs	Required information
Directly recoverable	NDC, NCC	Endpoint capacity channel
Same-cycle engineered	ICR, QTY	Raw cross-signal or time record
History-dependent	SOH, CFR	First or preceding cycles
Algebraically derived	CF	$1-\mathrm{SOH}$

Appendix

Table 4: Information requirements of the seven benchmark outputs. The categories describe target construction rather than statistical independence.

Hyperparameter	Value
Optimizer	AdamW
Learning rate / weight decay	$5\times 10^{-5}$ / $10^{-4}$
Maximum epochs	220
Batch size / accumulation	4 / 8
Dropout	0.2
Hidden dimension / default layers	64 / 2 where applicable

Appendix

Table 5: Default common training settings. Architecture-specific depths follow Table 2 and the corresponding reference implementations.

Component	Setting
Identity coefficient $\alpha$	0.90
Base operator-update scale $\eta_{0}$	0.05
Conditioned update scale $\eta_{K,b,t}$	$(0.0125,0.05)$
Frobenius perturbation scaling	$A/\max(1,\lVert A\rVert_{F})$
Lifted dimension / rank	96 / 8
Scalar modulation range $d_{b,t}$	$(0.90,1.10)$

Appendix

Table 6: Default full-model STEK settings. The conditioned lifted linear map and its consistency loss form the training-only Koopman-inspired regularization branch.

Task	$\boldsymbol{\mu_{j}^{\mathrm{train}}}$	$\boldsymbol{\sigma_{j}^{\mathrm{train}}}$
NDC	0.9311598	0.1476210
SOH	0.9813603	0.0279242
CF	0.0186397	0.0279242
CFR	$1.6988796\times 10^{-4}$	$3.3969091\times 10^{-4}$
ICR	85.042342	120.520454
NCC	0.9296390	0.1455620

Appendix

Table 7: Training-label means and standard deviations in original target units. NDC and NCC correspond to the implementation outputs named SOC and CA, respectively.

Masking probability $p$	Runs	Realized $p_{\mathrm{pred}}$	Overall	$\mathrm{Overall}_{-\mathrm{CFR}}$
0.00	3	0.000	0.1242	0.0796
0.25	30	0.264	0.1252	0.0807
0.50	30	0.500	0.1257	0.0813
0.75	30	0.748	0.1265	0.0821
1.00	3	1.000	0.1271	0.0829

Appendix

Table 8: Test-time availability stress test for prediction-relevant structural descriptors. The

dQ/dV

lifecycle terrain and endpoint-derived structural proxy are independently masked at the cell level with probability

p

. Realized

p_{\mathrm{pred}}

is computed over these two prediction-relevant descriptors. Material context is excluded because it is inactive in the inference pathway. Intermediate probabilities use ten mask seeds for each of three model seeds;

p=0

and

p=1

use one deterministic evaluation per model seed.

Source	Chemistry / form factor	Candidate cells	Effective cells	Test windows
CALCE	Li-ion, LCO/graphite, prismatic	13	13	65
HNEI	Li-ion, LCO+NCM442/graphite, 18650	14	14	70
MATR	Li-ion, LFP/graphite, 18650	169	169	845

Appendix

Table 9: Held-out sources and evaluable sample counts.

		TimesNet		DLinear		PatchTST
Task	Metric	+STEK	Base	+STEK	Base	+STEK	Base
NDC	MAE $\downarrow$	0.90 $\pm$ 0.06	0.99 $\pm$ 0.15	1.90 $\pm$ 0.09	2.55 $\pm$ 0.05	1.09 $\pm$ 0.04	1.14 $\pm$ 0.02
NDC	MaxAE $\downarrow$	7.65 $\pm$ 1.15	7.98 $\pm$ 1.38	44.76 $\pm$ 2.02	56.69 $\pm$ 0.83	7.83 $\pm$ 0.78	7.27 $\pm$ 1.44
SOH	MAE $\downarrow$	0.40 $\pm$ 0.03	0.47 $\pm$ 0.02	0.75 $\pm$ 0.03	1.03 $\pm$ 0.03	0.54 $\pm$ 0.06	0.57 $\pm$ 0.02
SOH	RMSE $\downarrow$	0.99 $\pm$ 0.02	1.12 $\pm$ 0.10	1.52 $\pm$ 0.06	1.77 $\pm$ 0.01	1.17 $\pm$ 0.04	1.23 $\pm$ 0.01
CF	MAE $\downarrow$	0.38 $\pm$ 0.02	0.45 $\pm$ 0.03	0.79 $\pm$ 0.03	1.03 $\pm$ 0.03	0.54 $\pm$ 0.05	0.56 $\pm$ 0.02

Appendix

Table 10: Generality of STEK across temporal backbones on MIX-battery. NDC, SOH, CF, NCC, and Overall are multiplied by

10^{2}

, while CFR is multiplied by

10^{4}

\downarrow

indicates that lower values are better. For each backbone, the better result between the base model and its STEK-enhanced variant is bolded.

Model	Params	FLOPs	Memory	Overall $\downarrow$
MambaLithium	0.09	0.02	14.93	$16.24\pm 0.59$
LiPM	0.66	0.12	13.64	$22.11\pm 1.10$
CPTransformer	0.12	0.01	17.63	$16.51\pm 1.29$
iTransformer	0.78	0.001	12.70	$15.08\pm 0.47$
Koopa	0.09	0.001	12.81	$16.27\pm 0.45$
KNF	0.23	0.01	10.64	$17.26\pm 0.38$

Appendix

Table 11: Model efficiency under the MIX-battery seven-output setting. Parameters, FLOPs, and memory are reported in M, G, and MB, respectively. Parameter counts include all trainable modules, including the training-only Koopman-inspired branch. FLOPs measure one FP16 prediction forward pass and exclude offline lifecycle-terrain construction, precomputed material-text embeddings, training losses, backward propagation, and optimizer updates. Values are reported as mean

\pm

standard deviation over three seeds.

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