Reinforcement Learning and Metaheuristics for Feynman Integral Reduction

TL;DR

Two new methods for optimizing the integration-by-parts (IBP) reduction of Feynman integrals are proposed, which involve an agent interacting with an environment in a step-by-step manner and learning the best actions to take given an observation of the environment.

Abstract

We propose new methods for optimizing the integration-by-parts (IBP) reduction of Feynman integrals, an important computational bottleneck in modern perturbative calculations in quantum field theory. Using the simple example of one-loop massive bubble integrals, we pose the problem of minimizing the number of arithmetic operations in reducing a target integral to master integrals via the Laporta algorithm. This is a nontrivial combinatorial optimization problem over the ordering of IBP equation generation (from pairs of seed integrals and IBP operators) and the ordering of integral elimination. Our first proposed method is reinforcement learning, which involves an agent interacting with an environment in a step-by-step manner and learning the best actions to take given an observation of the environment (in this case, the current state of the IBP reduction process). The second method is using metaheuristics, e.g. simulated annealing, to minimize the computational cost as a black-box function of numerical priority values that control the orderings. For large-scale problems, the number of free parameters can be compressed by using a small neural network to assign priority values. Remarkably, with almost no human guidance, both methods lead to IBP reduction schemes that are competitive with the most efficient human-designed algorithms. We also found interpretable features in the AI results that may be applicable to more complicated problems.

Figures (11)

• Figure 1: One-loop massive bubble integrals.
• Figure 2: Graphical representations of the state of the IBP reduction process for the target integral $I_{1,2}$ after steps (1a) and (2b).
• Figure 3: The $20$ candidate seed integrals (white squares) and $17$ additional auxiliary integrals (light gray squares on the peripherals).
• Figure 4: Cost in reducing $I_{3,3}$ to master integrals versus the number of episodes used for training, in an example run that reached the best cost 74 after 4280 episodes. The cost at the beginning of training is 1234, outside the plot range, but drops below 550 after 30 episodes.
• Figure 5: The optimal IBP reduction steps found by reinforcement learning, for reducing $I_{3,3}$ marked as the red square.