发表机构
The Hong Kong University of Science and Technology（香港科技大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对多尺度PDE粗粒化动力学，提出通量型时空神经算子，结合傅里叶卷积与因果时间算子，嵌入通量型归纳偏置，经三类算例验证可实现稳定高保真的长时程自回归预测。
AI 中文摘要
我们研究多尺度偏微分方程（PDE）系统中粗粒化动力学的数据驱动预测。采用无封闭项的算子学习视角，我们应用线性粗粒化映射，并直接从滤波后的高保真轨迹中学习已解析场的替代演化算子。受Mori-Zwanzig形式主义启发，我们提出一种时空神经算子，将Ω×[-T_in,0]上的已解析历史块映射到Ω×[0,T_out]上的已解析未来块。空间混合使用傅里叶卷积，而时间混合使用因果核算子，该算子在时间滞后上具有位置注意力权重。此因果时间算子编码已解析动力学中的有限记忆效应，同时保持历史到未来映射的方向性。为提高回推鲁棒性并抑制非保守伪影，我们通过以显式散度形式参数化窗口更新，嵌入通量型归纳偏置。我们还提供了一种数据驱动的指南，用于从滤波轨迹计算的封闭项注入诊断的去相关时间中选择记忆长度T_in。我们在粗粒化粘性Burgers方程、Kuramoto-Sivashinsky方程和二维湍流流动上进行验证，获得了稳定的自回归回推，具有改进的长时程精度和统计保真度。
英文摘要
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.
