Symbolic-Numeric Integration of Univariate Expressions based on Sparse Regression

TL;DR

This paper introduces a hybrid symbolic-numeric method for univariate indefinite integration, combining rule-based transformations with sparse regression to enhance integration capabilities in modern scientific computing environments.

Contribution

It presents a novel integration approach that integrates symbolic rule-based methods with sparse regression, tailored for modern scientific computing systems like Julia's SciML.

Findings

Successfully integrates a wide range of univariate expressions

Uses only a few dozen basic integration rules

Enhances symbolic integration in scientific computing ecosystems

Abstract

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.