Towards Intelligent Manufacturing: Spatio-Temporal Learning of Process–Material Dynamics with Attention-Driven Neural Operators

We present an uncertainty-aware Attention-based Spatio-Temporal Neural Operator (ASNO) framework for additive manufacturing. This work extends the original ASNO model by incorporating uncertainty quantification and generalization across material property space. Although ASNO can learn complex spatio-temporal dynamics in scientific and engineering systems, it does not provide predictive uncertainty, which is essential for reliability, safety, and interpretability in data-limited and high-stakes manufacturing settings. To address this limitation, we integrate a Laplace-based approximation to estimate epistemic uncertainty within operator learning. The proposed architecture follows an implicit–explicit structure inspired by the Backward Differentiation Formula, separating temporal evolution through a Transformer encoder and spatial interactions through a Nonlocal Attention Operator. This design enables stable, accurate, and computationally efficient forecasting of high-dimensional spatio-temporal fields. We evaluate ASNO across benchmark chaotic and PDE systems, including the Lorenz system, Darcy flow, and Navier–Stokes equations, demonstrating strong predictive performance with reliable uncertainty estimates, achieving 94% prediction interval coverage with narrow bounds. We further apply the framework to Directed Energy Deposition, where it accurately predicts full-field temperature distributions, generalizes across different alloy materials, and supports inference of melt pool characteristics. Beyond forward prediction, the learned operator and its uncertainty estimates provide a foundation for inverse identification and accelerated material discovery, such as estimating temperature-dependent thermal properties from sparse observations. Overall, this work advances uncertainty-aware surrogate modeling for intelligent manufacturing applications.