Comparing Machine Learning and Interpolation Methods for Loop-Level Calculations

TL;DR

This paper compares machine learning and interpolation methods for efficient function approximation in high-energy physics, finding neural networks outperform traditional techniques in higher dimensions for speed and accuracy.

Contribution

It provides a systematic comparison of four interpolation and three machine learning methods on complex physics functions, highlighting neural networks' advantages in high dimensions.

Findings

Radial Basis Function performs well in low dimensions.

Neural networks excel in higher dimensions for speed and accuracy.

Traditional methods struggle with curse of dimensionality.

Abstract

The need to approximate functions is ubiquitous in science, either due to empirical constraints or high computational cost of accessing the function. In high-energy physics, the precise computation of the scattering cross-section of a process requires the evaluation of computationally intensive integrals. A wide variety of methods in machine learning have been used to tackle this problem, but often the motivation of using one method over another is lacking. Comparing these methods is typically highly dependent on the problem at hand, so we specify to the case where we can evaluate the function a large number of times, after which quick and accurate evaluation can take place. We consider four interpolation and three machine learning techniques and compare their performance on three toy functions, the four-point scalar Passarino-Veltman $D_0$ function, and the two-loop self-energy master integral $M$. We find that in low dimensions ($d = 3$), traditional interpolation techniques like the Radial Basis Function perform very well, but in higher dimensions ($d=5, 6, 9$) we find that multi-layer perceptrons (a.k.a neural networks) do not suffer as much from the curse of dimensionality and provide the fastest and most accurate predictions.