We propose a scheme for data-driven parameterization of unresolved dimensions of dynamical systems based on the mathematical framework of quantum mechanics and Koopman operator theory. Given a system in which some components of the state are unknown, this method involves defining a surrogate system in a time-dependent quantum state which determines the fluxes from the unresolved degrees of freedom at each timestep. The quantum state is a density operator on a finite-dimensional Hilbert space of classical observables and evolves over time under an action induced by the Koopman operator. The quantum state also updates with new values of the resolved variables according to a quantum Bayes' law, implemented via an operator-valued feature map. Kernel methods are utilized to learn data-driven basis functions and represent quantum states, observables, and evolution operators as matrices. The resulting computational schemes are automatically positivity-preserving, aiding in the physical consistency of the parameterized system. We analyze the results of two different modalities of this methodology applied to the Lorenz 63 and Lorenz 96 multiscale systems, and show how this approach preserves important statistical and qualitative properties of the underlying chaotic systems.
翻译:我们根据量子力学和Koopman操作员理论的数学框架,提出了一个动态系统未解决维度的数据驱动参数化方案。鉴于这个系统的某些组成部分未知,这个方法涉及根据时间决定每个时间步骤未解决的自由度变化的量子状态确定代位系统。量子状态是一个在古典可观测的有限维度Hilbert空间上的密度操作器,并在Koopman操作员的推动下逐步演变。量子状态还根据量子贝斯法更新了已解决变量的新值,通过操作员估价地貌图实施。内尔方法用于学习数据驱动基函数,并代表量态、可观察性和进化操作器作为矩阵。由此产生的计算方法是自动保值,有助于参数化系统的物理一致性。我们分析了对Lorenz 63和Lorenz 96多尺度系统应用的两种不同方法的结果,并展示了这种方法如何保存基本混乱系统的重要统计和定性特性。