Quantitative Understanding of PDF Fits and their Uncertainties
Authors: Amedeo Chiefa, Luigi Del Debbio, Richard Kenway
Organizations: The Higgs Centre for Theoretical Physics, School of Physics and Astronomy, The University of Edinburgh, Peter Guthrie Tait Road, Edinburgh EH9 3FD, United Kingdom
Abstract
Parton Distribution Functions (PDFs) play a central role in describing experimental data at colliders and provide insight into the structure of nucleons. As the LHC enters an era of high-precision measurements, a robust PDF determination with a reliable uncertainty quantification has become mandatory in order to match the experimental precision. The NNPDF collaboration has pioneered the use of Machine Learning (ML) techniques for PDF determinations, using Neural Networks (NNs) to parametrise the unknown PDFs in a flexible and unbiased way. The NNs are then trained on experimental data by means of stochastic gradient descent algorithms. The statistical robustness of the results is validated by extensive closure tests using synthetic data. In this work, we develop a theoretical framework based on the Neural Tangent Kernel (NTK) to analyse the training dynamics of neural networks. This approach allows us to derive, under precise assumptions, an analytical description of the neural network evolution during training, enabling a quantitative understanding of the training process. Having an analytical handle on the training dynamics allows us to clarify the role of the NN architecture and the impact of the experimental data in a transparent way. Similarly, we are able to describe the evolution of the covariance of the NN output during training, providing a quantitative description of how uncertainties are propagated from the data to the fitted function. While our results are not a substitute for PDF fitting, they do provide a powerful diagnostic tool to assess the robustness of current fitting methodologies. Beyond its relevance for particle physics phenomenology, our analysis of PDF determinations provides a testbed to apply theoretical ideas about the learning process developed in the ML community.
We investigate a normalizing-flow approach for reconstructing parton distribution functions (PDFs) from synthetic matrix-element data. Our framework combines Gaussian Process priors with invertible neural networks to learn a posterior distribution over PDFs consistent with limited Ioffe-time data. We demonstrate that the architecture preserves physical constraints and extrapolation properties.
Neural networks (NNs) are inherently multidimensional classifiers that learn complex, non-linear relationships among input observables. While their flexibility enables unprecedented performance in high-energy physics (HEP) analyses, it also makes them sensitive to small variations in their inputs. Consequently, the propagation and estimation of systematic uncertainties in NN-based models remain an open challenge. There are indications that uncertainties derived in control regions or from nominal variations of input features can underestimate the true model uncertainty, potentially leaving biases unaccounted for. Inspired by insights from adversarial-attack studies in machine learning, we explore how subtle perturbations, fully consistent with the experimental uncertainties on the input observables, can lead to substantial changes in NN outputs, while keeping the one-dimensional and correlated input distributions nearly unchanged. Using a set of representative HEP tasks, including event classification and object identification, and testing across a variety of network architectures, we demonstrate that networks can be systematically "fooled" at significant rates within the allowed uncertainty envelopes. Building on this observation, we introduce a quantitative framework to probe and measure the hidden sensitivity of neural networks to realistic experimental variations, providing a practical path to evaluate and control their systematic uncertainty in physics analyses.
Lucie Flek, Philipp Alexander Jungs, Akbar Karimi +6
In wide neural networks, the Neural Tangent Kernel (NTK) remains approximately constant during training, providing a powerful theoretical tool for studying training dynamics, generalization, and connections to kernel methods. However, this theory is largely restricted to regression losses. It was previously thought that training on a classification loss, or more generally losses involving nonlinear output transformations, breaks this property, leading to divergent logits and a breakdown of the linearization. In this paper, we extend NTK theory to classification by identifying conditions under which wide neural networks remain in the lazy training regime. We show that parameter-space regularization ensures a constant NTK during training for cross-entropy loss, while in the absence of regularization the regime is recovered when targets are non-degenerate, i.e. when all classes have strictly positive probability. Under these conditions, training is well-approximated by the linearized model, yielding an explicit characterization of the solution in terms of the NTK. We further analyze the distribution of trained predictors induced by random initialization and relate this notion of model uncertainty to Bayesian methods.
Jonathan Plenk, Sergio Calvo-Ordonez, Alvaro Cartea +3