Uncovering Singularities in Feynman Integrals via Machine Learning

High-precision scattering amplitudes are crucial for testing the Standard Model at colliders and modeling gravitational waves from compact binaries. Upcoming experiments such as the HL-LHC, CEPC, FCC-ee, and third-generation gravitational-wave detectors will achieve unprecedented precision, demanding theoretical predictions of comparable accuracy, particularly in the form of accurate multi-loop scattering amplitudes. Around a decade ago, obtaining precise predictions for two-to-three particle collider processes beyond next-to-leading order was widely considered infeasible. This changed with advances in evaluating complicated two-loop Feynman integrals and interpreting them in terms of Chen's iterated integrals. Key steps include deriving and solving differential equations for master integrals and assembling full amplitudes, often with finite-field techniques. In this context, the concept of the symbol alphabet and associated function spaces has become central for multi-loop studies [1, 2]. These tools capture the algebraic structure of iterated integrals, first explored by Chen in the 1970s [3], which naturally arise in canonical-form differential equations [4] and can be expressed as nested d-log integrals.