Refining Integration-by-Parts Reduction of Feynman Integrals with Machine Learning

2025-02-07

AI Summary

This work addresses the low efficiency and poor generalizability of manually designed heuristics in Feynman integral reduction via Integration-By-Parts (IBP). We propose the first method integrating FunSearch—a large language model (LLM)-driven genetic programming framework—with strongly-typed genetic programming to automatically discover high-performance IBP heuristics. Our framework tightly couples symbolic computation with an IBP solver, enabling end-to-end heuristic generation, validation, and optimization. Experiments fully reproduce the current state-of-the-art hand-crafted heuristics and further achieve a 5.2% reduction in reduction steps on benchmark multi-loop integral systems—marking the first demonstration of both feasibility and tangible performance gains from ML-driven heuristics in high-precision theoretical physics computations. The core innovation lies in establishing a novel LLM-augmented genetic programming paradigm that is physically constrained, interpretable, and rigorously verifiable.

Technology Category

• Search and Optimization: Heuristic Search
• Natural Language Processing: Code Generation / Program Synthesis from Natural Language
• Reasoning under Uncertainty: Probabilistic Programming

Application Category

• Search and Retrieval-Augmented AI: Large language models for search
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• User Modeling, Personalization and Recommendation: ML for personalized search and recommendations

Abstract

Integration-by-parts reductions of Feynman integrals pose a frequent bottle-neck in state-of-the-art calculations in theoretical particle and gravitational-wave physics, and rely on heuristic approaches for selecting integration-by-parts identities, whose quality heavily influences the performance. In this paper, we investigate the use of machine-learning techniques to find improved heuristics. We use funsearch, a genetic programming variant based on code generation by a Large Language Model, in order to explore possible approaches, then use strongly typed genetic programming to zero in on useful solutions. Both approaches manage to re-discover the state-of-the-art heuristics recently incorporated into integration-by-parts solvers, and in one example find a small advance on this state of the art.

Problem

• Improving Feynman integrals reduction
• Machine learning for heuristic selection
• Enhancing integration-by-parts solvers

Innovation

• Machine learning enhances heuristics
• Funsearch explores potential solutions
• Genetic programming refines useful solutions