Neural Network Surrogate Modeling for Stochastic Finite Element Method Using Three-Dimensional Graph Representations: A Comparative Study

Abstract

Abstract. Modern engineering design increasingly relies on probabilistic simulation to account for uncertainties in geometry and loading conditions. The stochastic finite element method (SFEM) has become standard as a way to address this need, using thousands of deterministic FEM evaluations to estimate the uncertainty. However, this creates a prohibitively high computational cost that can inhibit efficient design exploration. Neural network (NN) surrogate models offer a promising alternative, shifting computation to a one-time upfront training cost while enabling near-instantaneous subsequent evaluations for iterative design tasks. However, effective NN surrogates for SFEM must learn to directly predict distributions that traditionally emerge from iterative sampling and aggregation of system responses across varying parameter spaces. Although previous research has explored various NN architectures for physical simulations, their effectiveness specifically for SFEM problems that combine geometric complexity with stochastic loading conditions, particularly in predicting converged distribution of physical fields that require understanding relationships between global and local features, remains inadequately addressed. This work addresses this gap by systematically evaluating 11 NN architectures organized into three distinct learning mechanism categories: attention-based, message passing, and hierarchical approaches. Our evaluation using 3D geometries with stochastic point elastic loading reveals that while these surrogate models achieve inference speeds orders of magnitude faster than traditional SFEM (milliseconds versus hours for traditional SFEM), their accuracy remains below the level required for full replacement of SFEM in iterative design applications. Our findings identify specific architectural trade-offs, highlighting avenues for hybrid approaches that may better balance computational efficiency with predictive accuracy.