Comparing Machine Learning and Interpolation Methods for Loop-Level Calculations

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.

30 pages, 17 figures, v2:added a few references, v3: new title, added a few references

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• GPflow: A Gaussian process library using TensorFlow
• Scalar one-loop 4-point integrals
• DART: Dropouts meet Multiple Additive Regression Trees
• NNLO QCD corrections to three-photon production at the LHC
• A factorisation-aware Matrix element emulator
• Optimising simulations for diphoton production at hadron colliders using amplitude neural networks
• Targeting Multi-Loop Integrals with Neural Networks
• Machine-learning interpolation of population-synthesis simulations to interpret gravitational-wave observations: a case study

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