Symbolic-numeric integration of univariate expressions based on sparse regression

Abstract

The majority of computer algebra systems (CAS) support symbolic integration using a combination of heuristic algebraic and rule-based (integration table) methods. In this paper, we present a hybrid (symbolic-numeric) method to calculate the indefinite integrals of univariate expressions. Our method is broadly similar to the Risch-Norman algorithm. 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 designed for numerical and machine learning applications. The symbolic part of our method is based on the combination of candidate terms generation (ansatz generation using a methodology borrowed from the Homotopy operators theory) combined with rule-based expression transformations provided by the underlying CAS. The numeric part uses sparse regression, a component of the Sparse Identification of Nonlinear Dynamics (SINDy) technique, to find the coefficients of the candidate terms. We show that this system can solve a large variety of common integration problems using only a few dozen basic integration rules.

Bridging HPC Communities through the Julia Programming Language

“…Working well with CSE applications is crucial for the success of Julia in HPC because these applications allow for testing proven technologies and algorithms at different scales with varying levels of support in a broad community. Success stories in different CSE fields include algebraic geometry (Breiding and Timme 2018), astronomy at petascale (Regier et al 2018), cancer therapies (Pich et al 2019), computer algebra and number theory (Fieker et al 2017), electrical engineering (Plietzsch et al 2022), epidemic modeling (Weitz et al 2020), high-performance geophysical simulations (Räss et al 2022), fluid dynamics (Ramadhan et al 2020;Ranocha et al 2022), semiconductor theory (Frost 2017), symbolic-numeric computing (Ketcheson and Ranocha 2021;Iravanian et al 2022;Ma et al 2021), quantum optics (Krämer et al 2018), quantum chemistry (Aroeira et al 2022), quantum physics (Herbst et al 2021), and many others.…”