Unstable periodic orbits (UPOs) are the non-chaotic, dynamical building blocks of spatio-temporal chaos, motivating a first-principles based theory for turbulence ever since the discovery of deterministic chaos. Despite their key role in the ergodic theory approach to fluid turbulence, identifying UPOs is challenging for two reasons: chaotic dynamics and the high-dimensionality of the spatial discretization. We address both issues at once by proposing a loop convergence algorithm for UPOs directly within a low-dimensional embedding of the chaotic attractor. The convergence algorithm circumvents time-integration, hence avoiding instabilities from exponential error amplification, and operates on a latent dynamics obtained by pulling back the physical equations using automatic differentiation through the learned embedding function. The interpretable latent dynamics is accurate in a statistical sense, and, crucially, the embedding preserves the internal structure of the attractor, which we demonstrate through an equivalence between the latent and physical UPOs of both a model PDE and the 2D Navier-Stokes equations. This allows us to exploit the collapse of high-dimensional dissipative systems onto a lower dimensional manifold, and identify UPOs in the low-dimensional embedding.

翻译：不稳定周期轨道（UPOs）是时空混沌的非混沌动力学构建模块，自确定性混沌发现以来，一直激励着基于第一性原理的湍流理论发展。尽管UPOs在流体湍流的遍历理论方法中具有关键作用，但其识别面临两大挑战：混沌动力学特性与空间离散化的高维性。我们通过提出一种直接在混沌吸引子的低维嵌入空间内运行的UPO循环收敛算法，同时解决了这两个问题。该收敛算法规避了时间积分，从而避免了指数级误差放大带来的不稳定性，并在通过自动微分技术沿学习到的嵌入函数回推物理方程所获得的潜动力学空间中进行操作。这种可解释的潜动力学在统计意义上是精确的，且关键在于嵌入过程保持了吸引子的内部结构——我们通过模型偏微分方程和二维纳维-斯托克斯方程的潜空间UPOs与物理空间UPOs的等价性证明了这一点。这使得我们能够利用高维耗散系统向低维流形的坍缩特性，在低维嵌入空间中识别UPOs。