Most computer algebra systems (CAS) support symbolic integration as core functionality. The majority of the integration packages use a combination of heuristic algebraic and rule-based (integration table) methods. In this paper, we present a hybrid (symbolic-numeric) methodology to calculate the indefinite integrals of univariate expressions. The primary motivation for this work is to add symbolic integration functionality to a modern CAS (the symbolic manipulation packages of SciML, the Scientific Machine Learning ecosystem of the Julia programming language), which is mainly designed toward numerical and machine learning applications and has a different set of features than traditional CAS. The symbolic part of our method is based on the combination of candidate terms generation (borrowed from the Homotopy operators theory) with rule-based expression transformations provided by the underlying CAS. The numeric part is based on sparse-regression, a component of Sparse Identification of Nonlinear Dynamics (SINDy) technique. We show that this system can solve a large variety of common integration problems using only a few dozen basic integration rules.
翻译:大多数计算机代数系统(CAS)支持象征性整合,作为核心功能。大多数集成包使用超常代数和基于规则(集成表)方法的组合。在本文中,我们提出了一个混合(Symbolic-numberic)方法,用以计算单体表达式的无限整体。这项工作的主要动机是将象征性整合功能添加到现代CAS(SciML的象征性操作包,SciML的SciML,Julia编程语言的科学机器学习生态系统)中,后者主要设计为数字和机器学习应用程序,具有与传统的CAS不同的一套特征。我们方法的象征部分是以候选术语生成(从Hoomatopty操作者理论中吸收)与基本化学文摘取的基于规则的表达转换相结合为基础。数字部分基于微弱反射,即非线性动态(SINDI)的简单识别技术的组成部分。我们显示,这个系统只能使用几十条基本集成规则解决大量共同融合问题。