Explainable AI-assisted Optimization for Feynman Integral Reduction

TL;DR

This work targets the computational bottleneck of Feynman integral reductions via IBP identities. It introduces a priority-function framework learned by FunSearch, a hybrid of large language models and genetic algorithms, to minimize the required seeding integrals and memory footprint. The approach yields substantial gains in memory and speed across one-loop and multi-loop planar and non-planar six-particle integral families, including dramatic reductions for complex subsectors (e.g., up to 3058x fewer seeds). By delivering an interpretable, scalable method, the paper demonstrates AI-assisted optimization as a practical tool for enabling higher-order perturbative calculations in high-energy physics.

Abstract

We present a novel approach to optimizing the reduction of Feynman integrals using integration-by-parts identities. By developing a priority function through the FunSearch algorithm, which combines large language models and genetic algorithms, we achieve significant improvements in memory usage and computational efficiency compared to traditional methods. Our approach demonstrates substantial reductions in the required seeding integrals, making previously intractable integrals more manageable. Tested on a variety of Feynman integrals, including one-loop and multi-loop cases with planar and non-planar configurations, our method demonstrates remarkable scalability and adaptability. For reductions of certain Feynman integrals with many dots and numerators, we observed an improvement by a factor of 3058 compared to traditional methods. This work provides a powerful and interpretable framework for optimizing IBP reductions, paving the way for more efficient and practical calculations in high-energy physics.

Figures (4)

• Figure 1: This flowchart outlines the operational sequence of the FunSearch algorithm, which is designed to optimize priority functions for IBP reduction tasks. The process begins with retrieving of one or two existing priority functions from the database. These functions, coupled with a concise explanation of the IBP reduction process, are formatted into a best-shot prompt. This prompt serves as input for a pre-trained LLM, which generates new priority functions based on the provided information. FunSearch then extracts these newly generated priority functions from the LLM's output. The extracted priority functions undergo a validation process to determine if they are executable. If successful, the evaluator utilizes these functions to perform IBP reductions, assessing their effectiveness. Functions that demonstrate superior performance are subsequently integrated back into the database, enriching it for future iterations of the algorithm. This cyclical process ensures continuous improvement and refinement of the priority functions used in IBP reductions.
• Figure 2: Visualization of the Laporta, Box, and Ellipse-like priority. The target integral $I(15,10)$ is marked as 'Target', and the master Integral is marked as 'Master'.
• Figure 3: Two six-particle phase-space integral families
• Figure 4: Evolution of the average seeding integral number with different hyperparameters. Each line represents a different experiment. Red dot for each line represents the best priority function achieved in this experiment.