Analytic Regression of Feynman Integrals from High-Precision Numerical Sampling

TL;DR

This paper introduces a method combining high-precision numerical sampling and analytic function space knowledge to deduce exact Feynman integrals, complementing traditional symbolic and top-down approaches.

Contribution

It presents a novel bottom-up technique for analytically reconstructing Feynman integrals from numerical data using lattice reduction and function space assumptions.

Findings

Effective in deducing exact integrals from high-precision data

Trade-offs identified between data points, precision, and computational effort

Applicable to a broad class of problems beyond Feynman integrals

Abstract

In mathematics or theoretical physics one is often interested in obtaining an exact analytic description of some data which can be produced, in principle, to arbitrary accuracy. For example, one might like to know the exact analytical form of a definite integral. Such problems are not well-suited to numerical symbolic regression, since typical numerical methods lead only to approximations. However, if one has some sense of the function space in which the analytic result should lie, it is possible to deduce the exact answer by judiciously sampling the data at a sufficient number of points with sufficient precision. We demonstrate how this can be done for the computation of Feynman integrals. We show that by combining high-precision numerical integration with analytic knowledge of the function space one can often deduce the exact answer using lattice reduction. A number of examples are given as well as an exploration of the trade-offs between number of datapoints, number of functional predicates, precision of the data, and compute. This method provides a bottom-up approach that neatly complements the top-down Landau-bootstrap approach of trying to constrain the exact answer using the analytic structure alone. Although we focus on the application to Feynman integrals, the techniques presented here are more general and could apply to a wide range of problems where an exact answer is needed and the function space is sufficiently well understood.