Empirical fits to inclusive electron-carbon scattering data obtained by deep-learning methods

TL;DR

This work develops two neural-network–based, model-independent parametrizations of inclusive electron–carbon scattering cross sections spanning from the quasielastic peak through resonance production to deep-inelastic scattering, with quantified uncertainties. Uncertainty estimation is achieved via bootstrap ensembles (Model A) and Monte Carlo dropout (Model B) within a likelihood framework that accounts for dataset normalization through a penalty in $\chi_{\text{tot}}$, yielding uncertainties around $7\%$. Model A shows superior interpolation and extrapolation capabilities compared with Model B, especially when predicting data outside the training domain, and predictions align with spectral-function calculations and DIS data. The resulting fits, including normalization handling, are suitable for integration into neutrino MC generators such as NuWro, and the authors release the fits publicly for broader use.

Abstract

Employing the neural network framework, we obtain empirical fits to the electron-scattering cross sections for carbon over a broad kinematic region, extending from the quasielastic peak through resonance excitation to the onset of deep-inelastic scattering. We consider two different methods of obtaining such model-independent parametrizations and the corresponding uncertainties: based on the bootstrap approach and the Monte Carlo dropout approach. In our analysis, the $χ^2$ defines the loss function, including point-to-point and normalization uncertainties for each independent set of measurements. Our statistical approaches lead to fits of comparable quality and similar uncertainties of the order of $7$%. To test these models, we compare their predictions to test datasets excluded from the training process and theoretical predictions obtained within the spectral function approach. The predictions of both models agree with experimental measurements and theoretical calculations. We also perform a comparison to a dataset lying beyond the covered kinematic region, and find that the bootstrap approach shows better interpolation and extrapolation abilities than the one based on the dropout algorithm.

Figures (9)

• Figure 1: Kinematic domain covered by the experimental data Arrington:1995hsArrington:1998psBagdasaryan:1988hpBaran:1988twBarreau:1983htDai:2018xhiDay:1993mdFomin:2010eiO'Connell:1987agSealock:1989nxWhitney:1974hr considered in our analysis, shown in the planes of (top) energy transfer and four-momentum transfer, and (bottom) energy transfer and cosine of the scattering angle.
• Figure 2: The neural network architecture for model A. The light gray box corresponds to the hidden layer of fully connected neurons with ReLU activation function. The dark gray box denotes the batch normalization layer. In the output, there is the sigmoid function. In the architecture for model B, not depicted, each fully connected layer is followed by the dropout layer. In both models A and B, the network has ten layers of hidden units, consisting of $300$ units each.
• Figure 3: Illustration of the algorithm used in model A to estimate the fit's uncertainty. Bootstrap datasets are generated from the original data according to the normal distribution (not depicted), and fits are performed $50$ times (green lines). The cross section's estimate corresponds to the mean of the fits (black line), and the uncertainty corresponds to their standard deviation (not shown). The data are taken from Ref. Barreau:1983ht.
• Figure 4: Histograms of the normalized residuals, $(d\sigma_i - d \sigma_i^\text{fit})/d\sigma_i$, for (top) model A and (bottom) model B. Test data are included.
• Figure 5: Histograms of the standard deviation normalized by the experimental central value of (top) model A and (bottom) model B. Test data are included.