An Attention-Based Spatio-Temporal Neural Operator with Uncertainty Quantification for Dynamical Systems

In this paper we present the Attention-based Spatio-Temporal Neural Operator (ASNO), an operator-learning architecture that decouples temporal evolution from spatial coupling. The design follows an implicit-explicit interpretation of Backward Differentiation Formula (BDF) integration: a time-series Transformer delivers explicit temporal extrapolation while a Nonlocal Attention Operator applies implicit spatial refinement. Epistemic uncertainty is estimated post hoc via a diagonal Linear Laplace Approximation with negligible overhead. Across Lorenz, Darcy, and two-dimensional incompressible Navier--Stokes systems, ASNO attains state-of-the-art or competitive accuracy under comparable parameter budgets, is resolution-agnostic, and maintains stable long-horizon rollouts, enabling uncertainty-aware modeling of high-dimensional fields.