Direct Interval Propagation Methods using Neural-Network Surrogates for Uncertainty Quantification in Physical Systems Surrogate Model
Interval uncertainty propagation typically requires solving expensive optimization problems for each input, making it infeasible for high-fidelity physics simulations. This paper proposes Direct Interval Propagation (DIP), reframing the task as interval-valued regression using neural surrogates to bypass optimization entirely. The authors extend DeepONet architectures to handle interval inputs and benchmark three distinct approaches---naive regression, bound propagation (IBP/CROWN), and interval neural networks---demonstrating orders-of-magnitude speedups on benchmark problems.
The paper presents a pragmatic approach to accelerating interval uncertainty quantification by replacing optimization loops with direct neural prediction. While the demonstrated speedups are compelling for the 1D regression case, the evaluation for PDEs relies on a strong monotonicity assumption that sidesteps the core challenge of non-monotonic interval propagation. The absence of the optimization baseline for PDE experiments limits the empirical validation of the central claim that these methods serve as viable surrogates for expensive propagation.
The speedup claims are substantiated in the 1D regression benchmark, where even the slowest DIP method achieves inference times approximately three orders of magnitude faster than optimization-based propagation (Table 3). The extension of DeepONet to interval-valued branch inputs via separate lower and upper bound latent representations (Equations 10-11) is technically sound and extends operator learning to uncertainty quantification. The comparative study across naive regression, verification-based bounds (IBP/CROWN), and interval arithmetic networks (INN) provides a useful taxonomy of available strategies.
The PDE experiments (Sections 4.1.2 and 4.1.3) assume monotonic behavior where "the lower and upper input bounds consistently yield $g_L$ and $g_U$," which avoids solving the actual optimization problem and limits applicability to real engineering systems with non-monotonic responses. Consequently, the optimization baseline (Opt-Prop) is excluded from PDE comparisons due to "prohibitive computational cost," leaving the speedup claims unvalidated on the full-field problems where they matter most. The data augmentation strategy for pointwise data is described but its effectiveness appears mixed, with the authors noting "difficulties that arise when training operator-learning models with augmented interval datasets."
The evidence supports the claims for simple regression, where RMSE and coverage metrics (PICP, PINAW) are reported against both the optimization baseline and ground-truth intervals (Tables 1-2). However, for PDEs, the comparison is limited to relative performance between DIP methods, with no ground-truth interval bounds obtained via optimization for validation. The timing comparisons between methods are confounded by implementation in different frameworks (PyTorch for IBP/CROWN, TensorFlow for Naive/INN), as noted in Table 6: "The IBP and CROWN-based implementations are not directly comparable to the Naive and INN approaches, as the former are implemented in PyTorch whereas the latter are implemented in TensorFlow."
The paper does not provide a code repository or supplementary implementation details. While architecture sizes are specified, critical hyperparameters such as learning rates, regularization coefficients $\lambda$, and training durations are reported as "tuned separately" without specific values in the main text, deferring details to an Appendix that is referenced but not fully reproduced here. The lack of publicly available code for the interval DeepONet implementation and the data augmentation pipeline presents a significant barrier to reproducing the PDE results.
In engineering, uncertainty propagation aims to characterise system outputs under uncertain inputs. For interval uncertainty, the goal is to determine output bounds given interval-valued inputs, which is critical for robust design optimisation and reliability analysis. However, standard interval propagation relies on solving optimisation problems that become computationally expensive for complex systems. Surrogate models alleviate this cost but typically replace only the evaluator within the optimisation loop, still requiring many inference calls. To overcome this limitation, we reformulate interval propagation as an interval-valued regression problem that directly predicts output bounds. We present a comprehensive study of neural network-based surrogate models, including multilayer perceptrons (MLPs) and deep operator networks (DeepONet), for this task. Three approaches are investigated: (i) naive interval propagation through standard architectures, (ii) bound propagation methods such as Interval Bound Propagation (IBP) and CROWN, and (iii) interval neural networks (INNs) with interval weights. Results show that these methods significantly improve computational efficiency over traditional optimisation-based approaches while maintaining accurate interval estimates. We further discuss practical limitations and open challenges in applying interval-based propagation methods.
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