Neural network-based spring element method using deep operator network for efficient uncertainty evaluation of pile-supported structures

Abstract Soil exhibits strong nonlinearity and spatial variability, significantly influencing the behavior of piles and pile-supported structures. The discrete spring element method (DSEM) captures soil–pile interaction (SPI) using p–y and t–z springs but requires dense discretization and repeated simulations when soil profiles vary, resulting in high computational cost. While machine learning (ML) provides efficient surrogate models for single-pile analysis, its direct application to large-scale pile-supported systems is hindered due to the curse of dimensionality. Hybrid Neural Networks Spring Element (NNSE) approaches alleviate this burden by embedding ML-based surrogates within the DSEM framework; however, conventional multilayer perceptron (MLP)-based NNSE models approximate parametric, case-specific mappings and therefore require retraining for different stratigraphic configurations, limiting their efficiency in stochastic analyses involving spatial variability. To overcome this limitation, this study proposes a Deep Operator Network-based NNSE framework (DeepONet-NNSE) that learns the nonlinear operator mapping between soil stiffness functions and pile load–displacement responses. By representing soil stratigraphy as a functional input, the trained DeepONet generalizes across unseen soil profiles without retraining, enabling efficient uncertainty propagation under random soil fields. Compared with conventional DSEM, the proposed approach reduces computational cost by approximately 60% while maintaining high predictive accuracy in demonstrated uncertainty evaluation of pile-supported structures considering soil spatial variability. The results highlight the potential of operator learning-based surrogates for designing large-scale pile-supported structures while accounting for nonlinear SPI and complex soil spatial variability.