Direct interval propagation methods using neural-network surrogates for uncertainty quantification in physical systems surrogate model
Published in Knowledge-Based Systems, 2026
In engineering applications, uncertainty propagation refers to the systematic characterisation of a system output under uncertain inputs. Specifically for interval uncertainty, the objective is to determine the lower and upper bounds of the output, given interval-valued inputs. Such uncertainty propagation plays a crucial role in engineering tasks, including robust design optimisation and reliability analysis, where accurate characterisation of uncertainty is essential for safe and reliable decision-making. However, standard interval propagation requires solving optimisation problems that can be computationally expensive, particularly when dealing with complex physical systems. To address this challenge, surrogate models have been developed to enable efficient interval propagation. Although surrogate models are computationally more efficient, standard surrogate-based approaches typically only replace the evaluator function within the optimisation loop, which still requires a large number of inference calls. Therefore, we propose to directly estimate the output interval by reframing the problem as an interval-valued regression task. In this work, we present a comprehensive study of strategies for direct interval propagation using NN-based surrogate models, including standard multilayer perceptrons (MLPs) and deep operator networks (DeepONet). We investigate and compare three distinct approaches: (i) naive interval propagation through standard architectures, (ii) bound propagation techniques such as Interval Bound Propagation (IBP) and CROWN, and (iii) interval neural networks (INNs) with interval weights. Our results demonstrate that these methods are significantly more efficient compared to traditional optimisation-based interval propagation and are able to provide accurate interval estimates. We also discuss the limitations and open challenges associated with implementing interval-based propagation in practice.