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.