论文标题
操作员代数的数据同化
Data Assimilation in Operator Algebras
论文作者
论文摘要
我们开发了一个代数框架，用于顺序数据吸收部分观察到的动态系统。在此框架中，贝叶斯数据同化嵌入了非亚伯运算符代数中，该代数可通过乘法运算符和密度运算符（量子状态）来表示可观察到的物体。在代数方法中，数据同化的预测步骤由动态系统的Koopman操作员引起的量子操作表示。此外，分析步骤是通过量子效应来描述的，量子效应概括了贝叶斯观察性更新规则。将此公式投影到有限维矩阵代数上，会导致（i）自动保留阳性的新计算数据同化方案； （ii）使用用于机器学习的内核方法来适应一致的数据驱动近似。此外，这些方法是在量子计算机上实施的自然候选者。在气候模型中，Lorenz 96多尺度系统和El Nino Southern振荡的数据同化的应用在预测技能和不确定性量化方面显示出令人鼓舞的结果。
We develop an algebraic framework for sequential data assimilation of partially observed dynamical systems. In this framework, Bayesian data assimilation is embedded in a non-abelian operator algebra, which provides a representation of observables by multiplication operators and probability densities by density operators (quantum states). In the algebraic approach, the forecast step of data assimilation is represented by a quantum operation induced by the Koopman operator of the dynamical system. Moreover, the analysis step is described by a quantum effect, which generalizes the Bayesian observational update rule. Projecting this formulation to finite-dimensional matrix algebras leads to new computational data assimilation schemes that are (i) automatically positivity-preserving; and (ii) amenable to consistent data-driven approximation using kernel methods for machine learning. Moreover, these methods are natural candidates for implementation on quantum computers. Applications to data assimilation of the Lorenz 96 multiscale system and the El Nino Southern Oscillation in a climate model show promising results in terms of forecast skill and uncertainty quantification.
