Koopman算子在非线性切换批式流加发酵建模中的应用

Applications of the Koopman operator in nonlinear switched fed-batch fermentationmodeling

摘要

带切换操作的微生物批式流加发酵合成1,3-丙二醇(1,3-PD)过程的建模依然面临巨大挑战,原因在于刻画该过程的动力系统具有强烈的非线性和不确定性。利用机器学习方法来构建此类模型已成为研究热点,但现有技术的可解释性仍是一大难题。近期,Koopman算子——一种能够沿着具有切换操作的非线性动力系统轨迹,控制特征函数演化的线性算子——被用于复杂动力学建模。本文提出一种基于可解释的Koopman算子建模方法,其核心优势在于能够为具有切换操作的非线性系统提供线性且无限维的描述。在该建模方法中,采用一种基于径向基函数(RBF)的扩展动态模态分解算法(EDMD-RBF)来获得Koopman算子的有限维近似,并对其收敛性进行了分析。数值算例显示,在相同数据条件下,采用反常S型径向基函数的EDMD-RBF算法,得到的线性受控系统在预测非线性受控系统时,精度明显优于其他径向基函数方案。

Abstract

Modeling the microbial fed-batch fermentation process with switching operations for 1,3-propanediol (1,3-PD) synthesis remains highly challenging, due to the strong nonlinearity and uncertainty inherent in its dynamic system. Using machine-learning approaches to build such models has become a hot topic, yet the interpretability of existing technologies remains a significant challenge. Recently, the Koopman operator—a linear, infinite-dimensional operator that governs the evolution of eigenfunctions along trajectories of nonlinear switched systems—has emerged as a powerful tool for describing complex dynamics. This paper propases an interpretable Koopman-based modeling framework whose key merit lies in providing a linear yet infinite-dimensional description of the switched nonlinear fermentation system. Within this framework an Extended Dynamic Mode Decomposition algorithm that uses radial basis functions (EDMD-RBF) is devised to furnish a finite-dimensional approximation of the Koopman operator, and its convergence properties are rigorously analyzed. Numerical experiments show that the linear controlled model produced by EDMD-RBF with an anomalous sigmoid RBF predicts the nonlinear controlled fermentation significantly more accurately than counterparts built with other RBF choices.

关键词

Koopman算子 / 切换操作 / 动态扩展模式-径向基函数 / 非线性受控动力系统

Key words

Koopman operator / switching operator / EDMD-RBF / nonlinear controlled dynamic system

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基金资助

辽宁省自然科学基金计划项目(2024-MS-015)