从合成湍流到真实解：一种深度扩散模型用于发现纳维-斯托克斯方程中的周期轨道
An abundance of periodic orbits in two-dimensional turbulence revealed by a diffusion-based generative model
AI总结：
本文提出一种深度扩散模型，用于发现纳维-斯托克斯方程中的周期轨道，通过生成合成轨迹并优化得到111个新周期轨道。
摘要：
生成式人工智能在合成模仿复杂现实系统的数据方面表现出显著的成功，但其在物理模型中发现数学上有意义的结构的潜力仍被低估。在本工作中，我们展示了如何利用生成扩散模型来发现非线性偏微分方程——湍流状态下二维纳维-斯托克斯方程的未知解。该模型在直接数值模拟湍流的数据上进行训练，学会生成类似物理合理轨迹的时间序列。通过仔细修改模型的时间结构并强制执行支配方程的对称性，我们生成了时间周期性的合成轨迹，尽管训练数据中没有包含周期性轨迹。这些合成轨迹随后通过迭代求解器进行优化，得到111个新的周期轨道（POs），具有非常短的周期。我们的结果揭示了该系统PO结构中之前未观察到的丰富性，并表明生成式人工智能在非线性动力系统复杂解空间中的作用：不是作为模拟和现有求解器的替代品，而是作为导航这些解空间的互补工具。
英文摘要：
Unstable periodic orbits offer a route to reconstructing the statistical properties of chaotic systems, including turbulence. Short periodic orbits are particularly important in periodic-orbit expansions, yet their abundance and diversity remain largely unexplored even in the extensively studied two-dimensional Kolmogorov flow. We search for short relative periodic orbits in this flow at $Re=40$, considering all distinct classes of recurrence symmetry. We train a generative diffusion model on nonperiodic vorticity sequences from a direct numerical simulation of turbulence, producing time series that resemble physically plausible trajectories. The model incorporates symmetry properties of the system but is otherwise trained solely on these data, without enforcing the governing equations. After training, we modify its architecture to generate synthetic trajectories that are periodic up to spatial symmetry transformations, without retraining its weights. These synthetic trajectories are then refined into exact solutions using an iterative solver, yielding 111 new relative periodic orbits with periods $T < 3$, including 40 with $T<2$. Rather than revealing a hierarchy dominated by a small number of short cycles, this abundance suggests that quantitatively converged periodic-orbit descriptions of spatiotemporal chaos may require substantially larger orbit libraries than anticipated.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
