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arxiv: 2605.11460 · v1 · submitted 2026-05-12 · 💻 cs.LG · cs.SY· eess.SY

Recognition: no theorem link

Beyond Prediction: Interval Neural Networks for Uncertainty-Aware System Identification

Mehmet Ali Ferah , Tufan Kumbasar
Authors on Pith no claims yet

Pith reviewed 2026-05-13 02:09 UTC · model grok-4.3

classification 💻 cs.LG cs.SYeess.SY
keywords interval neural networkssystem identificationprediction intervalsuncertainty quantificationLSTMNODEdynamical systemsinterval arithmetic
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The pith

Interval neural networks produce calibrated prediction intervals for dynamical systems by propagating uncertainty through interval arithmetic without probabilistic assumptions.

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

The paper develops interval versions of LSTM and NODE networks that treat parameters and activations as intervals rather than single numbers. Uncertainty is propagated forward using interval arithmetic rules, and two training approaches convert or jointly optimize these models for both accuracy and interval quality. Cascade INN first trains a standard network then converts it, while Joint INN optimizes both goals simultaneously with custom loss terms. Experiments across system identification benchmarks show the cascade version leads in point accuracy and the joint version leads in interval calibration and tightness. A new channel-wise elasticity measure is used to examine how uncertainty distributes across network channels under each strategy.

Core claim

Extending crisp neural networks to interval counterparts via interval arithmetic in LSTM and NODE architectures, combined with uncertainty-aware loss functions and parameterization tricks, enables the models to output prediction intervals that represent model uncertainty for system identification tasks; the joint training strategy yields better-calibrated intervals while the cascade strategy preserves superior point predictions.

What carries the argument

Interval Neural Networks (INNs) that replace scalar operations with interval arithmetic in LSTM and NODE layers, trained via either Cascade INN (two-stage conversion from crisp NN) or Joint INN (one-stage joint optimization of accuracy and interval precision) using uncertainty-aware losses.

Load-bearing premise

That propagating uncertainty through interval arithmetic in these architectures, together with the proposed losses and tricks, will produce intervals that accurately reflect true model uncertainty rather than merely over- or under-bounding outputs.

What would settle it

On a standard SysID benchmark dataset, if the J-INN prediction intervals fail to cover the observed outputs at a rate matching the target coverage level or if C-INN point predictions do not exceed the accuracy of ordinary crisp networks, the central claims would be refuted.

Figures

Figures reproduced from arXiv: 2605.11460 by Mehmet Ali Ferah, Tufan Kumbasar.

Figure 1
Figure 1. Figure 1: Snapshot of predicted system responses from Interval NODE (INODE) models [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Robot Arm Dataset: Performance Comparison for C-INN Strategy [PITH_FULL_IMAGE:figures/full_fig_p021_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Hair Dryer Dataset: Performance Comparison for C-INN Strategy [PITH_FULL_IMAGE:figures/full_fig_p021_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: MR-Damper Dataset: Performance Comparison for C-INN Strategy [PITH_FULL_IMAGE:figures/full_fig_p022_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Heat Exchanger Dataset: Performance Comparison for C-INN Strategy [PITH_FULL_IMAGE:figures/full_fig_p022_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Robot Arm Dataset: Performance Comparison for J-INN Strategy [PITH_FULL_IMAGE:figures/full_fig_p025_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Hair Dryer Dataset: Performance Comparison for J-INN Strategy [PITH_FULL_IMAGE:figures/full_fig_p025_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: MR-Damper Dataset: Performance Comparison for J-INN Strategy [PITH_FULL_IMAGE:figures/full_fig_p026_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Heat Exchanger Dataset: Performance Comparison for J-INN Strategy [PITH_FULL_IMAGE:figures/full_fig_p026_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Heatmaps of LPs resulting from the MR-Damper dataset using the J-INN strategy. [PITH_FULL_IMAGE:figures/full_fig_p029_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Channel-wise elasticity of models obtained for the MR-Damper Dataset [PITH_FULL_IMAGE:figures/full_fig_p030_11.png] view at source ↗
read the original abstract

System identification (SysID) is critical for modeling dynamical systems from experimental data, yet traditional approaches often fail to capture nonlinear behaviors. While deep learning offers powerful tools for modeling such dynamics, incorporating uncertainty quantification is essential to ensure reliable predictions. This paper presents a systematic framework for constructing and training interval Neural Networks (INNs) for uncertainty-aware SysID. By extending crisp neural networks into interval counterparts, we develop Interval LSTM and NODE models that propagate uncertainty through interval arithmetic without probabilistic assumptions. This design allows them to represent uncertainty and produce prediction intervals. For training, we propose two strategies: Cascade INN (C-INN), a two-stage approach converting a trained crisp NN into an INN, and Joint INN (J-INN), a one-stage framework jointly optimizing prediction accuracy and interval precision. Both strategies employ uncertainty-aware loss functions and parameterization tricks to ensure reliable learning. Comprehensive experiments on multiple SysID datasets demonstrate the effectiveness of both approaches and benchmark their performance against well-established uncertainty-aware baselines: C-INN achieves superior point prediction accuracy, whereas J-INN yields more accurate and better-calibrated prediction intervals. Furthermore, to reveal how uncertainty is represented across model parameters, the concept of channel-wise elasticity is introduced, which is used to identify distinct patterns across the two training strategies. The results of this study demonstrate that the proposed framework effectively integrates deep learning with uncertainty-aware modeling.

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

3 major / 2 minor

Summary. The paper proposes a framework for Interval Neural Networks (INNs) for uncertainty-aware system identification, extending LSTM and Neural ODE architectures via interval arithmetic to generate prediction intervals without probabilistic assumptions. It introduces Cascade INN (C-INN), a two-stage method that first trains a crisp network then converts it, and Joint INN (J-INN), a one-stage joint optimization of accuracy and interval quality using uncertainty-aware losses and parameterization tricks. Experiments on multiple SysID datasets claim C-INN achieves superior point prediction accuracy while J-INN produces more accurate and better-calibrated intervals than baselines; a channel-wise elasticity metric is introduced to analyze uncertainty representation across parameters.

Significance. If validated, the work provides a practical non-probabilistic alternative for uncertainty quantification in deep dynamical models, useful for control and safety-critical applications where calibrated bounds matter. The distinction between C-INN and J-INN training strategies offers actionable guidance, and the elasticity metric adds interpretability value. Strengths include the use of interval arithmetic for direct propagation and benchmarking against established baselines, though empirical grounding is limited by missing protocol details.

major comments (3)
  1. [Methods (interval LSTM propagation) and Experiments (calibration results)] The claim that J-INN yields better-calibrated intervals (abstract and experimental results) rests on empirical coverage metrics, but the manuscript provides no analysis of the dependency problem in interval arithmetic when applied to recurrent LSTM structures; correlated hidden states are treated independently, which can cause exponential interval widening over sequence length and produce conservative rather than calibrated bounds. No formal bound or correction via the proposed losses or two-stage vs. one-stage training is shown.
  2. [Experiments section] Superiority claims for C-INN point accuracy and J-INN interval calibration lack error bars, full data splits, hyperparameter search details, and ablation on the uncertainty-aware losses/parameterization tricks. Without these, it is impossible to rule out post-hoc selection or overfitting to the specific SysID datasets, undermining the central benchmarking conclusions.
  3. [Section introducing channel-wise elasticity] The channel-wise elasticity metric is introduced to reveal uncertainty patterns, but its definition and computation are not shown to be independent of the interval widths produced by the arithmetic; if elasticity is derived directly from the same interval outputs, it risks being tautological rather than providing new insight into model uncertainty.
minor comments (2)
  1. Notation for interval bounds (e.g., lower/upper) should be standardized across equations and figures for clarity.
  2. [Experiments] The abstract mentions 'well-established uncertainty-aware baselines' but the manuscript should explicitly list them with citations in the experimental setup.

Simulated Author's Rebuttal

3 responses · 1 unresolved

We thank the referee for the constructive and detailed feedback. We address each major comment below with honest clarifications and indicate where revisions will be made to improve the manuscript.

read point-by-point responses

  1. Referee: [Methods (interval LSTM propagation) and Experiments (calibration results)] The claim that J-INN yields better-calibrated intervals (abstract and experimental results) rests on empirical coverage metrics, but the manuscript provides no analysis of the dependency problem in interval arithmetic when applied to recurrent LSTM structures; correlated hidden states are treated independently, which can cause exponential interval widening over sequence length and produce conservative rather than calibrated bounds. No formal bound or correction via the proposed losses or two-stage vs. one-stage training is shown.

    Authors: We acknowledge the dependency problem in interval arithmetic for recurrent models as a known challenge, where independent treatment of correlated states can lead to interval overestimation. The J-INN joint optimization with uncertainty-aware losses is intended to produce tighter, better-calibrated intervals in practice, as shown by the empirical coverage results outperforming baselines. However, we do not provide a formal bound or theoretical correction for the widening effect. In revision we will add a limitations discussion section addressing this issue, describing how the training strategies empirically control widening on the evaluated datasets, and noting it as an open direction for future research. revision: partial

  2. Referee: [Experiments section] Superiority claims for C-INN point accuracy and J-INN interval calibration lack error bars, full data splits, hyperparameter search details, and ablation on the uncertainty-aware losses/parameterization tricks. Without these, it is impossible to rule out post-hoc selection or overfitting to the specific SysID datasets, undermining the central benchmarking conclusions.

    Authors: We agree these experimental details are required to substantiate the benchmarking claims and ensure reproducibility. In the revised manuscript we will add error bars computed over multiple independent runs, explicit descriptions of the train/validation/test splits for all datasets, the full hyperparameter search procedure and ranges explored, and ablation studies on the uncertainty-aware losses and parameterization tricks. These changes will directly address concerns about post-hoc selection or dataset-specific overfitting. revision: yes

  3. Referee: [Section introducing channel-wise elasticity] The channel-wise elasticity metric is introduced to reveal uncertainty patterns, but its definition and computation are not shown to be independent of the interval widths produced by the arithmetic; if elasticity is derived directly from the same interval outputs, it risks being tautological rather than providing new insight into model uncertainty.

    Authors: Channel-wise elasticity is defined as a normalized sensitivity measure: the relative change in interval width per channel under controlled input perturbations, allowing comparison of uncertainty distribution patterns between C-INN and J-INN. While computed from interval outputs, it yields comparative insights (e.g., concentrated vs. diffuse uncertainty across channels) not directly visible from raw widths or aggregate calibration metrics. We will revise the section to include the precise mathematical definition, computation procedure, and illustrative examples demonstrating its added interpretive value beyond tautology. revision: partial

standing simulated objections not resolved
  • Formal theoretical bound or correction for the interval dependency problem in recurrent LSTM structures.

Circularity Check

0 steps flagged

No significant circularity in the INN framework derivation

full rationale

The paper's core derivation extends crisp neural networks to interval counterparts via standard interval arithmetic applied to LSTM and NODE architectures, then defines two training procedures (C-INN as two-stage conversion and J-INN as joint optimization) along with uncertainty-aware losses and a channel-wise elasticity metric. None of these steps reduce by construction to fitted inputs or self-referential definitions; interval propagation follows established rules independent of the target prediction intervals, and the performance claims rest on external empirical benchmarks against uncertainty-aware baselines on SysID datasets. No load-bearing self-citations, imported uniqueness theorems, or smuggled ansatzes appear in the derivation chain. The framework is self-contained with independent experimental validation.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides insufficient detail to enumerate specific free parameters or axioms; the framework rests on the domain assumption that interval arithmetic suffices for uncertainty propagation in recurrent and continuous-time neural models. No invented entities beyond the INN construct itself are introduced.

pith-pipeline@v0.9.0 · 5552 in / 1071 out tokens · 62679 ms · 2026-05-13T02:09:19.784545+00:00 · methodology

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