基于累积量截断的伴随混沌
Adjoint chaos via cumulant truncation
摘要：
我们描述了一个简单而系统的方法，从混沌动力系统中获得近似的灵敏度信息，使用一个层次的累积量方程。由此产生的前向和伴随系统产生关于系统的泛函的梯度的信息，并且不遭受与混沌系统的切线线性表示相关联的收敛问题。我们关注的泛函是整体平均的量，其动态不一定是混沌的，因此我们分析系统的统计状态动态，而不是单个轨迹。该方法的目的是提取参数的敏感性信息，可以从直接数值模拟或实验获得的详细统计。我们提倡一种数据驱动的方法，将系统的累积量的观察，以确定一个最佳的封闭层次的累积量，不需要规范的模型参数。虽然从所得到的代理模型的灵敏度信息是近似的，该方法被设计用于湍流的分析，其自由度和复杂性的数量目前禁止使用更准确的技术。在这里，我们应用该方法从Rayleigh-Bénard对流的低维表示中获得泛函梯度。
英文摘要：
We describe a simple and systematic method for obtaining approximate sensitivity information from a chaotic dynamical system using a hierarchy of cumulant equations. The resulting forward and adjoint systems yield information about gradients of functionals of the system and do not suffer from the convergence issues that are associated with the tangent linear representation of chaotic systems. The functionals on which we focus are ensemble-averaged quantities, whose dynamics are not necessarily chaotic; hence we analyse the system's statistical state dynamics, rather than individual trajectories. The approach is designed for extracting parameter sensitivity information from the detailed statistics that can be obtained from direct numerical simulation or experiments. We advocate a data-driven approach that incorporates observations of a system's cumulants to determine an optimal closure for a hierarchy of cumulants that does not require the specification of model parameters. Whilst the sensitivity information from the resulting surrogate model is approximate, the approach is designed to be used in the analysis of turbulence, whose number of degrees of freedom and complexity currently prohibits the use of more accurate techniques. Here we apply the method to obtain functional gradients from low-dimensional representations of Rayleigh-Bénard convection.
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