TIDE: Trustworthy and Interpretable Battery Degradation Estimation with Contextual Learning and Symbolic Distillation

Elisa Y. M. Ang; Fang Liu; Jiawei Li; Peng Cheng Wang; Sumei Sun; Wei Zhang; Wen Yang Tan

REVIEW 4 major objections 4 minor 18 references

TIDE: Trustworthy and Interpretable Battery Degradation Estimation with Contextual Learning and Symbolic Distillation

T0 review · 4 major / 4 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read A battery health estimator that pairs accuracy with zero aging violations, made readable through symbolic distillation.

desk verdict Sensible structured SoH estimator with a credible accuracy story, but the trustworthiness headline is weaker than it looks because the MVR protocol never exercises the contextual component. read the letter →

arxiv 2607.14640 v1 pith:ZPQBNKTL submitted 2026-07-16 cs.LG

Wen Yang Tan , Jiawei Li , Fang Liu , Wei Zhang , Sumei Sun , Peng Cheng Wang , Elisa Y. M. Ang This is my paper

classification cs.LG

keywords batterystateofhealthSoHestimationtrustworthymachinelearningmonotoneresidualKolmogorov-ArnoldnetworkssymbolicdistillationagingconsistencyinterpretableML

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 tries to show that battery state-of-health estimation does not have to trade accuracy for trustworthiness and interpretability. Its proposed estimator, TIDE, combines a knowledge-driven degradation prior, a residual component that is non-increasing by construction in cycle index and internal resistance, and a flexible Kolmogorov-Arnold residual for battery-specific context. On a large public fast-charging battery dataset, TIDE reports the best overall fit (R²=0.964) and lowest per-battery spread, while exhibiting zero monotonicity violations—unlike accurate baselines that violate aging behavior in over 30% of perturbations. The trained model is then distilled into a compact symbolic expression that reproduces its behavior and reveals patterns such as late-life degradation acceleration. If true, this is a template for ML models that embed physical constraints rather than discovering them from data alone.

What carries the argument

The load-bearing object is a three-component additive backbone: C1 = a·exp(−b·i) − d·r, an exponential aging prior with trainable coefficients; C2, a sum of softplus terms with all effective weights constrained non-negative and a leading minus sign, guaranteeing a non-increasing correction; and C3, a Kolmogorov-Arnold network variant (MultKAN) with explicit spline activations for contextual residuals. The non-negativity constraint is what carries the trustworthiness claim: it builds monotonicity into the architecture rather than into the loss. Symbolic distillation then compresses the trained backbone into a compact mathematical expression, giving model-level interpretability.

What would settle it

Compute measured SoH as a function of cycle index and internal resistance on a dataset that includes rest periods, low-temperature cycles, or partial charging, and look for increases in true SoH as cycle index or resistance rises. If such non-monotone segments exist, refit TIDE and unconstrained baselines on them: the falsifying observation would be TIDE's errors concentrating on those segments, with larger deviation from measured SoH than the unconstrained model—showing the monotonicity constraint, not the physics, caused the 'zero violation.'

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Extended reading notes

Core claim

The central claim is that a structured additive decomposition of battery degradation—an exponential aging prior, a guaranteed-monotone residual, and an unconstrained context component—can deliver accuracy, trustworthiness, and interpretability simultaneously. The monotone residual is built from softplus activations with non-negative weights and a leading negative sign, so it cannot increase with cycle index or internal resistance by construction; this is the mechanism that yields zero aging-consistency violations. The contextual Kolmogorov-Arnold component is what supplies most of the accuracy: an ablation shows R² jumps from 0.318 to 0.896 when it is added, and the full model reaches R²=0.9

Load-bearing premise

The argument assumes SoH is genuinely non-increasing in cycle index and internal resistance under all operating conditions; if legitimate capacity recovery or resistance fluctuations violate that monotonicity, TIDE's zero-violation trustworthiness guarantee is partly a product of the constraint rather than of physical fidelity.

Editorial extensions

If this is right

SoH estimates from TIDE never predict a higher state of health when cycle index or internal resistance rises, holding other inputs fixed—this is guaranteed by architecture, not by post-hoc correction.
A vanilla Kolmogorov-Arnold network alone is not accurate enough (R²=0.572); the structured degradation prior is what makes a KAN-based residual viable.
The contextual component is the main driver of accuracy: without it, the model's R² collapses to 0.318, showing that battery health depends on operating conditions beyond simple aging trends.
Symbolic distillation works better from a structured teacher: TIDE's surrogate has lower error than surrogates distilled from comparable hybrid neural baselines despite similar teacher accuracy.
The final symbolic surrogate is compact enough to serve as a lightweight, human-readable estimator for battery management systems.

Reading between the lines

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

If monotonic aging is the right prior, the same decomposition could be transferred to other health indicators—remaining useful life, state of charge, or thermal state—where a physics-derived monotonicity constraint exists; the paper does not test this.
The zero-violation result depends on treating cycle index and internal resistance as the only aging directions; any dataset with capacity recovery or temperature-driven resistance drops would put the trustworthiness criterion itself to the test.
The symbolic surrogate's explicit form could be compiled directly into a battery management system or used to generate synthetic degradation data—an engineering consequence the paper only gestures at.
A natural testable extension is to compare TIDE's constrained estimates against unconstrained models on batteries with known capacity-recovery events; if errors concentrate exactly at those events, monotonicity is the bottleneck.

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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

Punchline: the three-component backbone is a sensible combination and the accuracy comparison is mostly credible, but the trustworthiness headline is weaker than it looks. The zero-MVR result is partly guaranteed by architecture and the evaluation protocol never exercises the contextual component.

What's new: the specific combination of an exponential knowledge prior, a structurally monotone residual, and a KAN residual, followed by symbolic distillation. Each block is known, but the integration for same-cycle SoH estimation is new, and the ablation shows the decomposition works: C3 drives the accuracy, C2 provides the aging-consistent correction, and the distilled surrogate in Eq. (7) is a compact and useful artifact. The comparison to CNN-BiGRU and CNN-BiLSTM is fair, and TIDE comes out with comparable RMSE, better R2, and lower variance across batteries.

Soft spots: First, the MVR protocol. It perturbs i or r_i while holding everything else fixed, including derived features like x_ir = i*r_i and rolling resistance that C3 uses. So C3's response to realistic co-varying aging is never tested. Since C2 is monotone by construction, the zero MVR mainly reflects C1+C2, not the full estimator. The paper needs to either perturb the derived features consistently or report MVR under joint aging. Second, Table II shows C1+C2 with lower RMSE than C1 but much lower R2—that is impossible pointwise on a fixed test set with standard formulas; the authors need to fix or explain the aggregation. Third, no code, no seeds or confidence intervals, which matters for an empirical paper. Fourth, the 19.7% improvement is an average of relative R2 gains over all baselines, dominated by the weak ones; the strongest baseline is essentially tied on RMSE. The abstract's phrasing overstates this.

None of these refute the core architecture. The accuracy story holds up, the interpretability artifacts are real, and the paper is clearly written. But the 'trustworthiness' pillar needs to be reworked or honestly relabeled as an architectural constraint rather than an empirically discovered property.

Recommendation: Send to peer review. It deserves referee time, with expectation of major revision on the MVR protocol, the ablation table, and code release. I'd also bring it to a reading group—it's a good case study in why monotonicity constraints and evaluation protocols need to match.

Referee Report

4 major / 4 minor

Summary. The paper proposes TIDE, a three-component backbone for battery state-of-health (SoH) estimation from BMS cycle-level features. The backbone combines a knowledge-guided exponential prior C1, a monotone residual C2 built from non-negative softplus weights, and an unconstrained MultKAN contextual residual C3, trained end-to-end and then distilled into a compact symbolic surrogate. The empirical claims are: (i) TIDE matches or improves on six baselines in RMSE/R2 on a battery-disjoint split of MIT-Stanford data; (ii) it achieves zero monotonic-violation rate (MVR) under cycle-index and internal-resistance perturbations; and (iii) the structured backbone and symbolic surrogate provide interpretable component- and model-level descriptions of aging behavior. The paper reports an average 19.7% improvement in R2 over baselines and emphasizes that accuracy, trustworthiness, and interpretability are jointly achieved.

Significance. If the claims hold, the paper makes a useful architectural contribution: separating a physically motivated monotone correction from a flexible KAN residual is a credible way to keep SoH estimates consistent with an aging prior while retaining accuracy, and symbolic distillation from a structured teacher is a sensible route to compact interpretability. The use of a public dataset with a battery-disjoint split, explicit feature-leakage exclusions, and per-battery RMSE standard deviation are good experimental practices. However, the strength of the contribution is limited by two gaps. First, the headline trustworthiness result (zero MVR) is largely a consequence of the monotonicity constraints and of the perturbation protocol's treatment of derived features, so it does not independently validate aging consistency. Second, a key ablation row in Table II is internally anomalous and undermines the stated contribution of the monotone component. The paper would be stronger with clearer diagnostics, multi-run variability, and a corrected/complete symbolic expression. The central architecture is defensible, but the evidence as presented requires revision.

major comments (4)

[§III.B.2 / Table III] The MVR diagnostic as defined holds 'remaining input features are held fixed' while perturbing i or r_i. But C3 and the symbolic surrogate use derived features that are functions of i and r_i, notably x_ir = i·r_i and rolling-resistance features. Holding these fixed while perturbing i or r_i means C3's strongest nonlinear pathway (Fig. 3c) is never exercised, so the zero-MVR result in Table III tests only part of the estimator. The claim in the Abstract and Conclusion that TIDE 'eliminates aging-consistency violations' is therefore stronger than the evidence. Please either include co-varying derived features in the perturbation protocol or explicitly state that the zero-MVR result is a partial monotonicity guarantee enforced by C1/C2, not a property of the full model on realistic aging trajectories.
[Table II / §III.B.1] The C1 and C1+C2 rows report RMSE 0.0397 vs 0.0356 but R2 0.318 vs 0.194. Under the stated convention that both metrics are computed per battery and averaged, average R2 is a nonlinear function of per-battery RMSE and target variance, so this is not formally impossible; however, such a large drop in average R2 despite lower average RMSE requires explanation. If R2 is pooled rather than per-battery averaged, the text should say so. As reported, the ablation narrative that C2 'helps minimize estimation error' while reducing R2 is ambiguous and load-bearing for the paper's component-contribution claim. Please clarify the exact metric computation and report the per-battery distributions.
[§II.C.1 / §II.C.2 / Table III] The zero-MVR results for C1, C1+C2, and TIDE are substantially by construction. C2 is non-increasing because of the non-negative softplus weights and the leading negative sign in Eq. (4). C1's monotonicity is not structurally guaranteed because a, b, d in Eq. (2) are unconstrained trainable coefficients; unless the training losses L_cyc/L_ir enforce monotonicity, the zero MVR for C1 is empirical and should be reported as such. The paper should separate 'enforced by architecture' from 'observed on data' and discuss whether the monotone-decline prior is valid under capacity-recovery or temperature transients. A concrete test would be to evaluate TIDE on batteries or cycles with transient SoH increases.
[Eq. (7) / §III.B.4] The symbolic surrogate expression is not well defined as typeset. Since s_v, x_ir, and \bar{r} are non-negative over realistic ranges, the term G(s_v, x_ir, \bar{r}) = s_v(-0.00794 x_ir - 0.02017 \bar{r} - 0.04988) is negative. If the square-root notation is taken literally, \hat{S} is complex and cannot reproduce the reported surrogate RMSE of 0.0129. The 'nested square-root' wording suggests a typesetting corruption. Please provide a corrected, evaluable expression, define all symbols including c, and verify it against Table IV.

minor comments (4)

[§III.A] No random seeds, confidence intervals, or multiple-run results are reported for the main comparisons. Given the small performance differences between TIDE and CNN-BiGRU (RMSE 0.0081 vs 0.0080), single-run results make the accuracy advantage hard to assess. At a minimum, report mean and standard deviation over several seeds.
[§III.B.2] The MVR protocol does not specify the perturbation step size, the number of perturbed samples per battery, or how the 0% rate is aggregated. Please define these details so the diagnostic is reproducible and comparable.
[Eq. (4)] The notation P and H is used without giving concrete values or an example. Since the monotone component is central to the trustworthiness claim, please state the selected features (x_mon) and the chosen P,H values.
[Fig. 4] The 'short/medium/long' lifespan curves appear to be single-battery examples with no confidence bands. Clarify whether these are representative individual batteries or aggregated means, and add uncertainty information where possible.

Circularity Check

1 steps flagged · score 6.0 of 10

Zero-MVR trustworthiness result is largely enforced: C2 is non-increasing by Eq. (4) and the MVR protocol freezes C3, so the headline aging-consistency claim is not independently tested.

self definitional [Section II.C.2 (Eq. 4) and Section III.A.2/Table III (MVR protocol and result)]
"Since softplus(·) is non-decreasing, the effective weights are non-negative, and Eq. (4) adopts a leading negative sign, so C2 is guaranteed to be non-increasing with respect to each selected feature. ... MVR, which measures the proportion of perturbations for which the predicted SoH increases when either i or r_i is increased while the remaining input features are held fixed. ... TIDE achieves zero MVR for both i and r_i, resulting in an overall MVR of 0.0%."

The final estimate (Eq. 5) is C1(i,r_i)+T^{-1}(C2(x_mon)+C3(x_ctx)), where x_ctx excludes the monotone features. Under the MVR protocol, perturbing i or r_i leaves all 'remaining input features' fixed, so C3's output is constant and the only structural monotone term is C2. Eq. (4) makes C2 non-increasing by construction via non-negative softplus weights and a leading negative sign. Hence the reported 0.0% MVR is a theorem about the model architecture and the evaluation protocol, not an empirical test of the full estimator under realistic co-varying aging. The trustworthiness headline is therefore largely self-definitional.

full rationale

The accuracy claim is independently grounded: the model is evaluated on a battery-disjoint held-out split (70/15/15) against standard baselines, so the 19.7% fidelity improvement is not circular. The symbolic surrogate is openly described as a PySR fit to the trained teacher, so its 'insights' are descriptive summaries of the fitted surrogate rather than independent first-principles discoveries; this limits their evidentiary weight but is not itself circular. The only significant circularity is in the trustworthiness pillar: Table III's 0.0% MVR follows from the definition of C2 (Eq. 4) plus the MVR protocol that holds remaining features fixed, thereby freezing C3. Because C3 is not exercised under co-varying aging features, the zero-violation result does not demonstrate that the complete TIDE estimator is aging-consistent. C1's unconstrained coefficients mean the zero is not formally guaranteed under every initialization, but the structural guarantee of C2 plus the frozen C3 is the operative mechanism. Self-citations in the references are not used as load-bearing evidence or uniqueness theorems, so they do not contribute to circularity. Overall, one central claimed result reduces partly by construction; the other two pillars retain independent content.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

No new physical entities are introduced. The model's postulates are the monotonic aging assumption, the exponential degradation prior, and dataset representativeness. The coefficients of C1/C2/C3 and the loss weights are fitted quantities; the numerical values are not reported, making the contribution hard to audit.

free parameters (5)

C1 coefficients a, b, d = not reported
Trainable coefficients in Eq. (2); they set the coarse degradation baseline and are fit to training data. Sign constraints are not stated, so monotonicity is not guaranteed.
C2 monotone residual weights (alpha, beta, gamma, beta0) = not reported
Trainable parameters in Eq. (4); non-negativity is enforced via softplus, which by construction makes C2 non-increasing in selected features.
C3 MultKAN parameters = not reported
Trainable KAN spline parameters in the contextual residual component; no monotonic constraints.
Loss weights lambda_cyc, lambda_ir, lambda_range, lambda_aux = not reported
Hyperparameters in Eq. (6) chosen to balance accuracy and constraints; values are never given, affecting reproducibility.
Symbolic surrogate expression (Eq. 7) = Eq. (7)
Selected by PySR to balance fidelity/complexity; the chosen expression is a fitted surrogate of the teacher, not a derived law.

assumptions (5)

domain assumption Battery SoH is non-increasing with cycle index and internal resistance
Defines trustworthiness/MVR in Section II.C.2 and Section III.A.2; capacity recovery and resistance noise can violate this in real cells.
ad hoc to paper The exponential form C1(i,r)=a exp(-b i)-d r captures the dominant aging trend
Eq. (2) is posited as empirical degradation knowledge; no electrochemical derivation or justification of functional form is provided.
domain assumption Cycle index and internal resistance are sufficient aging indicators from the BMS
Used in Eq. (2)-(4); other aging descriptors (e.g., SEI growth, lithium plating) are not considered.
domain assumption The MIT-Stanford fast-charging dataset with a battery-disjoint split represents practical deployment conditions
Section III.A; single chemistry/protocol; generalization to other conditions is assumed, not tested.
domain assumption Target-derived variables and future information have been fully excluded from the 34 features
Section III.A; leakage would inflate R2; the feature list and preprocessing are not fully enumerated.

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Cite this review

Pith. "Pith review of TIDE: Trustworthy and Interpretable Battery Degradation Estimation with Contextual Learning and Symbolic Distillation." pith.science (2026). https://pith.science/paper/ZPQBNKTL

@misc{pith2026260714640,
  author       = {Pith},
  title        = {Pith review of: TIDE: Trustworthy and Interpretable Battery Degradation Estimation with Contextual Learning and Symbolic Distillation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZPQBNKTL}},
  note         = {Machine review of arXiv:2607.14640}
}

read the original abstract

Battery health estimation is fundamental for battery management in battery-powered systems, where inaccurate health states may affect control, maintenance, and service life. It becomes even more critical in intelligent connected systems, where estimation errors can propagate across interconnected devices and downstream decisions. In this paper, we propose TIDE, a trustworthy and interpretable battery degradation estimator for reliable battery health estimation. TIDE jointly considers accuracy, trustworthiness, and interpretability, which are all essential for practical deployment and downstream decision making. To realize these objectives, TIDE combines battery-domain knowledge with operational measurements in a three-component backbone. A knowledge-guided degradation prior promotes trustworthy estimation, a monotone residual component provides interpretable aging-consistent refinement, and a contextual learning component captures battery-specific operational effects for improved accuracy. The trained backbone is then distilled into a compact symbolic surrogate that provides a concise model-level interpretation of its learned estimation logic. Experiments show that TIDE achieves strong estimation accuracy, improving overall estimation fidelity by an average of 19.7% over representative baselines. Its knowledge-guided prior and monotone residual modelling substantially reduce aging-consistency violations, supporting trustworthy estimation. Meanwhile, the backbone enables component-level interpretation, while symbolic distillation provides a compact model-level representation of the learned estimation logic. These results support the practical use of TIDE for battery health monitoring and decision support in intelligent connected systems.

Figures

Figures reproduced from arXiv: 2607.14640 by the authors.

**Figure 1.** Overview of the proposed TIDE framework. Degradation knowledge and BMS sensing features are jointly processed by a structured three-component backbone, followed by symbolic distillation to obtain a compact symbolic surrogate. information, they do not by themselves guarantee accurate, trustworthy, and interpretable estimation. To this end, we in the second layer introduce a structured backbone estimator consisting … view at source ↗

**Figure 2.** Learned activation structures of the TIDE backbone. Curve blocks denote learned KAN edge functions. Fig. 2a: C2 combines cycle index and internal resistance for aging-related correction. Fig. 2b: C3 models multiple operational features. Solid (dashed) paths indicate stronger (weaker) contributions. heterogeneous features contribute to its backbone. To further understand the quantitative influence of individual featu… view at source ↗

**Figure 3.** Representative learned response curves of T [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Representative physical insights revealed by the symbolic surrogate. Fig. 4a: Cycle-aging patterns across batteries [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

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