Hybrid Symbolic-Numeric and Numerically-Assisted Symbolic Integration

Most computer algebra systems (CAS) support symbolic integration using either algebraic or heuristic methods. This paper presents HYINT, a hybrid (symbolic-numeric) method to calculate the indefinite integrals of univariate expressions. Like the Risch-Norman algorithm, the symbolic part of HYINT generates an ansatz constituted of multiple candidate terms generated in parallel. The ansatz generator uses a combination of table lookup of integration rules and algebraic manipulations. The numeric part filters the candidate terms over the complex field and applies sparse regression, a component of the Sparse Identification of Nonlinear Dynamics (SINDy) technique, to find the coefficients of the terms in the ansatz. HYINT covers a larger range of potential integrals compared to the Risch-Norman algorithm. Moreover, the form of the final integral is similar to the integrand and consistent with what the users expect. 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. We show that this system can solve many common integration problems using only a few dozen basic integration rules. We also discuss numerically-assisted symbolic integration, where HYINT acts as an ansatz generator for other symbolic integration packages.