Quantitative Understanding of PDF Fits and their Uncertainties

TL;DR

This work reframes PDF fitting as an inverse problem analyzed through the Neural Tangent Kernel, enabling an analytic description of neural network training in the lazy regime via the flow solution f_t = U(t) f_0 + V(t) Y. By examining NTK initialization, training evolution, and eigenstructure, it shows how architecture and data steer learnable directions and propagate uncertainty, and provides a decomposition of the trained PDF covariance into initial-condition and data-driven contributions. The analytical framework is validated with simplified closure tests (L0/L1/L2) and cross-checked against numerical training, revealing how a frozen NTK captures kernel-learning aspects and how bias and variance evolve with training time. The approach offers a principled diagnostic for PDF uncertainties and suggests directions to extend kernel-based analyses to more realistic global fits and multi-flavor PDF determinations, potentially clarifying differences between fitting methodologies.

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.

Figures (28)

• Figure 1: Sampled distribution of a selected weight as a function of the number of replicas. The red line represents the underlying Gaussian distribution from which the weights are drawn. As the number of replicas is increased the distribution of the weight converges to the expected Gaussian.
• Figure 2: The empirical (left) and analytical (right) covariance matrices of the first, second and output layers of the NNPDF architecture (top to bottom). The covariance in the left panel is computed "bootstrapping" over an ensemble of replicas, initialised using the Glorot normal distribution. The covariance in the right panel is obtained by solving Eq. \ref{['eq:RecursionForK']} numerically. In order to reduce the bootstrap errors in the empirical covariance, an ensemble of 1000 replicas has been used for this figure.
• Figure 3: Relative difference between the empirical kernel, computed from an ensemble of networks at initialisation, and the recursive kernel obtained by iterating Eq. \ref{['eq:RecursionForK']} for the three layers of the NNPDF architecture. An ensemble of 1000 replicas has been used to reduce the bootstrap errors in the empirical covariance.
• Figure 4: Sampled distribution of the output $xT_3$ at $x=0.0065$ for two different ensemble sizes, $N_{\rm rep}=100$ (top) and $N_{\rm rep}=1000$ (bottom). Each column shows the distribution for a different network architecture, the latter displayed in the top left corner of each panel. The red line the represents the predicted Gaussian distribution as dictated by the kernel recursion formula in Eq. \ref{['eq:RecursionForK']}.
• Figure 5: The output of the ensemble of neural networks at initialisation using the NNPDF architecture in linear (left) and logarithm (right) scale. We compare the case of linear input $f(x)$ (blue) and the case of scaled input $f(x, \log x)$ (orange). The solid lines represent the mean value computed over an ensemble of 100 replicas, while the shaded bands represent the one-sigma uncertainty computed as the variance over the same ensemble. In the figure, we show $xT_3$ as used in the following sections.