Parameterized Machine Learning for High-Energy Physics

TL;DR

The paper addresses the need for discriminants in high-energy physics that remain effective across a continuous space of physics parameters (e.g., particle mass). It introduces a parameterized neural network $f(ar{x},\theta)$ that takes both event features and physics parameters as input, enabling smooth interpolation across parameter values. Through a toy example, a 1D $t\bar{t}$ resonance search, and a high-dimensional feature study, the authors demonstrate that the parameterized model can match or outperform fixed-$\theta$ networks and generalize to unseen $\theta$, while reducing training complexity. This approach offers practical benefits for physics analyses, including improved discriminants and better handling of nuisance parameters via integration with statistical tools like profile likelihoods.

Abstract

We investigate a new structure for machine learning classifiers applied to problems in high-energy physics by expanding the inputs to include not only measured features but also physics parameters. The physics parameters represent a smoothly varying learning task, and the resulting parameterized classifier can smoothly interpolate between them and replace sets of classifiers trained at individual values. This simplifies the training process and gives improved performance at intermediate values, even for complex problems requiring deep learning. Applications include tools parameterized in terms of theoretical model parameters, such as the mass of a particle, which allow for a single network to provide improved discrimination across a range of masses. This concept is simple to implement and allows for optimized interpolatable results.

Figures (8)

• Figure 1: Left, individual networks with input features $(x_1,x_2)$, each trained with examples with a single value of some parameter $\theta=\theta_a,\theta_b$. The individual networks are purely functions of the input features. Performance for intermediate values of $\theta$ is not optimal nor does it necessarily vary smoothly between the networks. Right, a single network trained with input features $(x_1,x_2)$ as well as an input parameter $\theta$; such a network is trained with examples at several values of the parameter $\theta$.
• Figure 2: Top, training samples in which the signal is drawn from a Gaussian and the background is uniform. Bottom, neural network response as a function of the value of the input feature $x$, for various choices of the input parameter $\theta$; note that the single parameterized network has seen no training examples for $\theta=-1.5,-0.5,0.5,1.5$.
• Figure 3: Feynman diagrams showing the production and decay of the hypothetical particle $X\rightarrow t\bar{t}$, as well as the dominant standard model background process of top quark pair production. In both cases, the $t\bar{t}$ pair decay to a single charged lepton ($\ell$), a neutrino ($\nu$) and several quarks ($q,b$).
• Figure 4: Top, distributions of neural network input $m_{WWbb}$ for the background and two signal cases. Bottom, ROC curves for individual fixed networks as well as the parameterized network evaluated at the true mass, but trained only at other masses.
• Figure 5: Distributions of some of the low-level event features for the decay of $X\rightarrow t\bar{t}$ with two choices of $m_X$ as well as the dominant background process.