The proton structure one mode at a time: a Neural Tangent Kernel analysis of a DIS fit

We study the training dynamics of a feed-forward neural network (NN) through the analysis of the neural tangent kernel (NTK) in a phenomenologically relevant fit of parton distribution functions (PDFs) from deep inelastic scattering (DIS) data. The exact integration of the gradient flow derived in a previous single-flavour analysis is generalised to the multi-flavour case, yielding an analytical perspective on the relationship between data and methodology during training. The Shapley value recently used in the context of PDFs is derived within our statistical framework for observables that depend linearly on the PDFs, and we outline a geometric interpretation of training. We expose the role of gradient-based optimisation algorithms and of the parametrisation in the regularisation of the ill-posed inverse problem that affects the determination of PDFs from experimental data, demonstrating that the regularisation amounts to the incremental inclusion of NTK modes, each learned at the timescale set by the corresponding eigenvalue. We find that only a few leading modes of the NTK are necessary to reconstruct the target PDFs, while the noise of the data is carried by the sub-leading ones learned at later stages of training. Finally, we consider the impact of the initialisation on the training dynamics, and recover a transient behaviour observed in the previous analysis and in the machine learning (ML) literature. Beyond sharpening our understanding of the training dynamics, our framework can be applied to other parametrisations (such as fixed functional forms) and ML applications in physics, as well as to stress tests in the presence of inconsistent data.