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.

翻译：我们提出采用统计回归作为投影算子，以实现Mori-Zwanzig形式体系中算子的数据驱动学习。我们提出了一种基于原则的方法，能够为任意回归模型提取马尔可夫算子与记忆算子。研究表明，线性回归的选择会得到一种近期提出的基于Mori投影算子的数据驱动学习算法，该算法本质上是高阶近似Koopman学习方法。同时我们发现，更具表达力的非线性回归模型能够自然地填补高度理想化且计算高效的Mori投影算子与最优但计算不可行的Zwanzig投影算子之间的空白。我们通过数值实验，针对一系列基于回归的投影方法（包括线性回归、多项式回归、样条回归和神经网络回归）提取了相应算子，结果表明随着回归模型复杂度的提升，性能呈现渐进式改善。本提案为提取记忆依赖修正项提供了通用框架，可直接应用于文献中多种针对平稳动力系统的数据驱动学习方法。