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

• URL: http://arxiv.org/abs/2608.18148v1
• Date: Tue, 11 Aug 2026 00:55:16 GMT
• Title: Flux-form spatiotemporal neural operators for coarse-grained dynamics of multiscale PDEs
• Abstract summary: We study data-driven prediction of coarse-grained dynamics in multiscale PDE systems.<n>Adopting a closure-free operator-learning viewpoint, we apply a linear coarse-graining map and learn a surrogate evolution operator for the resolved field.<n>We validate on the coarse-grained Burgers' equation, the Kuramoto-Sivashinsky equation, and two-dimensional turbulent flows.
• Score: 6.160113667162026
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• Abstract: We study data-driven prediction of coarse-grained dynamics in multiscale PDE systems. Adopting a closure-free operator-learning viewpoint, we 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, we propose a spatiotemporal neural operator mapping a resolved history slab on $Ω\times[-T_{\mathrm{in}},0]$ to a resolved future slab on $Ω\times[0,T_{\mathrm{out}}]$. Spatial mixing uses Fourier convolution, while temporal mixing uses a causal kernel operator with position-attention weights on time lags. This causal temporal operator encodes finite-memory effects in the resolved dynamics while preserving the directionality of the history-to-future map. To improve rollout robustness and suppress nonconservative artifacts, we embed a flux-form inductive bias by parameterizing the windowed update in explicit divergence form. We also provide a data-driven guideline for selecting the memory length $T_{\mathrm{in}}$ via the decorrelation time of a closure-injection diagnostic computed from filtered trajectories. We validate on the coarse-grained viscous Burgers' equation, the Kuramoto-Sivashinsky equation, and two-dimensional turbulent flows, obtaining stable autoregressive rollouts with improved long-horizon accuracy and statistical fidelity.

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