论文标题
基于回归的学习莫里兹万齐格操作员的预测
Regression-based projection for learning Mori-Zwanzig operators
论文作者
论文摘要
我们建议采用统计回归作为投影操作员，以使数据驱动以数据为基础的学习者 - Zwanzig形式主义中的操作员。我们提出了一种原则方法，用于为任何回归模型提取Markov和内存操作员。我们表明，线性回归的选择会导致基于Mori的投影操作员最近提出的数据驱动的学习算法，该算法是一种高阶近似Koopman学习方法。我们表明，更具表现力的非线性回归模型自然填补了高度理想化和计算有效的MORI投影操作员与最佳计算机上最佳的Zwanzig投影操作员之间的差距。我们进行了数值实验，并提取了一系列基于回归的投影的运算符，包括线性，多项式，样条和基于神经网络的回归，随着回归模型的复杂性的增加而显示出渐进的改进。我们的命题提供了一个通用框架来提取与内存有关的校正，并且可以轻松地应用于文献中固定动力学系统的一系列数据驱动的学习方法。
We propose to adopt statistical regression as the projection operator to enable data-driven learning of the operators in the Mori--Zwanzig formalism. We present a principled method to extract the Markov and memory operators for any regression models. We show that the choice of linear regression results in a recently proposed data-driven learning algorithm based on Mori's projection operator, which is a higher-order approximate Koopman learning method. We show that more expressive nonlinear regression models naturally fill in the gap between the highly idealized and computationally efficient Mori's projection operator and the most optimal yet computationally infeasible Zwanzig's projection operator. We performed numerical experiments and extracted the operators for an array of regression-based projections, including linear, polynomial, spline, and neural-network-based regressions, showing a progressive improvement as the complexity of the regression model increased. Our proposition provides a general framework to extract memory-dependent corrections and can be readily applied to an array of data-driven learning methods for stationary dynamical systems in the literature.
