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Direct Interval Propagation Methods using Neural-Network Surrogates for Uncertainty Quantification in Physical Systems Surrogate Model

Ghifari Adam Faza, Jolan Wauters, Fabio Cuzzolin, Hans Hallez, David Moens

Abstract

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.

Direct Interval Propagation Methods using Neural-Network Surrogates for Uncertainty Quantification in Physical Systems Surrogate Model

Abstract

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.
Paper Structure (27 sections, 38 equations, 9 figures, 17 tables, 2 algorithms)

This paper contains 27 sections, 38 equations, 9 figures, 17 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Classical interval propagation methods
  3. Direct interval propagation
  4. Naive direct interval propagation
  5. Standard implementation
  6. Naive direct interval propagation DeepONet implementation
  7. Bound propagation method
  8. IBP
  9. CROWN
  10. Bound propagation DeepONet implementation
  11. Interval neural-network method
  12. INN standard implementation
  13. INN DeepONet implementation
  14. Connection to interval estimation methods
  15. Experiments
  16. ...and 12 more sections

Figures (9)

  • Figure 1: Regularised artificial neural network (RANN) architecture for naive direct interval propagation.
  • Figure 2: Architecture illustration for (a) Standard DeepONet architecture. (b) Modified DeepONet architecture with interval-valued branch net. The standard architecture use standard feedforward neural network (FNN) on both branches, meanwhile the modified network replance FNN with the Naive interval architecture.
  • Figure 3: Interval neural network (INN) with DeepONet architecture.
  • Figure 4: Ideal interval dataset for a one-dimensional regression problem. Single rectangle represent a single pair of interval-valued $x$ and $y$.
  • Figure 5: Illustration of an ideal interval dataset for a one-dimensional PDE. The red curve denotes the input function with its interval uncertainty (dashed lines), while the blue curve denotes the corresponding PDE solution with its propagated interval uncertainty.
  • ...and 4 more figures