Flux-form spatiotemporal neural operators for coarse-grained dynamics of multiscale PDEs

Summary

This study addresses data-driven prediction of coarse-grained dynamics in multiscale PDE systems. Adopting a closure-free operator-learning viewpoint, the authors apply a linear coarse-graining map and learn a surrogate evolution operator for the resolved field directly from filtered high-fidelity trajectories. Motivated by the Mori-Zwanzig formalism, they propose a spatiotemporal neural operator mapping a resolved history slab to a resolved future slab. Spatial mixing uses Fourier convolution, while temporal mixing uses a causal kernel operator with position-attention weights on time lags. A flux-form inductive bias embeds explicit divergence form to improve rollout robustness and suppress nonconservative artifacts. Validation on viscous Burgers' equation, Kuramoto-Sivashinsky equation, and two-dimensional turbulent flows yields stable autoregressive rollouts with improved long-horizon accuracy and statistical fidelity.