aiqbench analysis
This research advances operator learning by integrating flux-form inductive biases into spatiotemporal neural operators to accurately predict coarse-grained dynamics in multiscale PDEs. By leveraging Fourier convolution for spatial mixing and causal attention for temporal lags, the method achieves stable long-horizon rollouts on chaotic systems like turbulent flows while preserving statistical fidelity. The work offers a rigorous framework for data-driven modeling of complex physical systems without relying on closure approximations.
Flux-form spatiotemporal neural operators for coarse-grained dynamics of multiscale PDEs
arXiv:2608.18148v1 Announce Type: cross 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 $\Omega\times[-T_{\mathrm{in}},0]$ to a resolved future slab on $\Omega\times[0,T_{\mathrm{out}}]$. 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.
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 $\Omega\times[-T_{\mathrm{in}},0]$ to a resolved future slab on $\Omega\times[0,T_{\mathrm{out}}]$. 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.
Chaotic Dynamics (nlin.CD); Machine Learning (cs.LG); Analysis of PDEs (math.AP); Computational Physics (physics.comp-ph)
arXiv:2608.18148 [nlin.CD]
(or arXiv:2608.18148v1 [nlin.CD] for this version)
https://doi.org/10.48550/arXiv.2608.18148 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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