Machine-learning-based neutrino-event generation constrained by experimental measurements

MODEL OPTIMIZATION

• The section expands the model-optimization scheme shown in Fig. 1 and describes its steps in detail.The expanded scheme is presented in Fig. S1.Source: MODEL OPTIMIZATION · p. 6

CROSS-SECTION DNN: PARAMETRIZATION AND TRAINING

• The σ-DNN supplies the cross section used to weight GAN-generated events and predicts interval averages for evaluating flux-averaged measurement bins.The section describes its tabulated training data, input and target parametrization, architecture, and optimization.Source: CROSS-SECTION DNN: PARAMETRIZATION AND… · p. 6

A. Training data and interval averaging

• Cross-section values are tabulated with NuWro from 0 to 20000 MeV in 5 MeV steps, with below-threshold values set to zero.The training data comprise the tabulated cross sections.Source: A. Training data and interval averaging · pp. 6–7
• For each interval, two endpoint indices are selected at random and the cross section is averaged over all tabulated values within it.The analysis uses 50 MeV intervals; the average assumes a uniform flux, and the single-energy prediction is the special case n = m.Source: A. Training data and interval averaging · p. 7

B. Input and target parametrization

• The network is trained on a bounded target formed by dividing the interval-averaged cross section by the interval energy, then recovers the physical cross section by inverting Eq. S-II.4.Energy inputs are rescaled to the unit interval.Source: B. Input and target parametrization · p. 7
• Training intervals are restricted to widths below 1 GeV, with Eν,m − Eν,n ≤ 1000 MeV.Source: B. Input and target parametrization · p. 7

C. Architecture

• The seven dense blocks contain 512, 256, 128, 64, 32, 16, and 8 neurons, followed by a single softplus-activated output.The softplus lower bound guarantees a non-negative predicted cross section.Source: C. Architecture · p. 7

D. Optimization

• The optimization avoids zero targets and uses a target-normalized loss so large-cross-section regions do not dominate, retaining fit quality near threshold.Training uses Nadam at learning rate 0.001 and mini-batches of 1000; the single-energy prediction is validated against the NuWro table in Fig. S2.Source: D. Optimization · pp. 7–8, Figure S2 · p. 8

• The generator produces fake samples that the discriminator learns to distinguish from real training samples, while the generator learns to fool the discriminator.Source: GENERATIVE ADVERSARIAL NETWORK:… · p. 8
• At convergence, the generator produces samples indistinguishable from the real ones.Source: GENERATIVE ADVERSARIAL NETWORK:… · p. 8
• A conditional GAN is used to reproduce event kinematics as a function of neutrino energy Eν.GANs for νµ scattering and domain adaptation via transfer learning were discussed in Refs. [18].Source: GENERATIVE ADVERSARIAL NETWORK:… · p. 8

A. Implementation and building blocks

• The generator maps a latent vector and rescaled Eν to muon kinematics, while the discriminator evaluates kinematics together with Eν.The generator uses three 100-neuron blocks; the discriminator uses six 141-neuron blocks.Source: A. Implementation and building blocks · p. 8

B. Variable parametrization

• The parametrization empirically flattens distributions near phase-space boundaries and constrains outputs to physically valid events, especially at Eν < 1 GeV.It keeps the muon energy between its rest mass and the incoming neutrino energy, reducing the need for large generator weights while allowing the discriminator to resolve distribution tails.Source: B. Variable parametrization · p. 9

C. Dense skip connections

• Dense skip connections make each block’s output available to all subsequent blocks, whose inputs are averaged before processing.The authors use these connections to ease information flow because adversarial training is inherently unstable.Source: C. Dense skip connections · p. 9, Figure S3 · p. 9

D. Training

• Base training uses NuWro events with matched real and generated Eν spectra; fine-tuning reweights Eν using the current σ-DNN and updates only final network layers.The σ-DNN is partially frozen, with only its final nine Keras layers trainable during fine-tuning.Source: D. Training · p. 9, §2 · p. 11

2. Drawing the reservoir

• The one-million-event reservoir samples truth-level NuWro events to match flux-weighted occupancies inferred from the MicroBooNE measurement, using absolute weights where unfolded central values are negative.The reservoir is a data-informed regularization sample, not a direct experimental target; the quantitative fit occurs separately.Source: §1 · p. 10, A. Reservoir construction and the… · p. 10, §2 · p. 11

3. Role of the reservoir

• Because the reservoir contains truth-level events but derives occupancies from a smeared measurement, it is not a truth-level estimate of the data and does not invert Ac.The authors note that inversion would be ill-posed given the regularized unfolding.Source: §3 · p. 11, §2 · p. 11
• Without the reservoir, the χ2 c fit alone distorts QE and ∆-resonance structure that smearing and coarse binning render invisible; the reservoir supplies that structure while χ2 c performs the data fit.Source: §3 · p. 12

1. Flux subdivision and the flux-averaged cross section

• Flux bins are subdivided so each measurement region contains whole sub-bins, enabling equivalent flux-averaged cross-section expressions from σ-DNN predictions or generated-event fractions.Source: §1 · p. 12, Figure S5 · p. 12

2. Differentiable histogram

• A smooth sigmoid-based histogram replaces hard bin assignment, allowing gradients to propagate from the binned loss through event positions to the generator.At finite sharpness, events near bin edges can be shared between neighboring bins; the approximation is compared with exact binning in Fig. S6.Source: §2 · pp. 12–13, Figure S6 · p. 13

3. From histogram to smeared prediction

• The soft histogram is cross-section weighted, forward-smeared by Ac, and compared with the signed measurement using a full-covariance χ2 c objective.The comparison takes place in measurement space and constrains both shape and normalization.Source: §3 · pp. 11, 13, §5 · p. 14

4. Gradient flow and the role of the σ-DNN

• The σ-DNN’s flux-averaged cross-section output rescales the histogram and receives gradients, while its role in sampling Eν is fixed and carries no gradient.Source: §4 · p. 13

• Figure S7 compares base and tuned models with MicroBooNE two-dimensional differential cross-section data across Eν bins.Source: Figure S7 · p. 14, C. Training results · p. 14
• The right panel shows (MC−Data)/Unc, with gray bands indicating measurement uncertainties and dashed vertical lines separating Eν regions.Source: Figure S7 · p. 14
• The black ratio-panel plot marks Data−|Data| to highlight negative cross-section values in the measurement.Source: Figure S7 · p. 14

COMPARISON WITH OUT-OF-TRAINING MICROBOONE MEASUREMENTS

• Table S1 reports χ2_c and χ2_v over the number of bins for base and tuned DNN models across three cross sections.The table caption names σ(Eν), dσ/dEµ, and dσ/dEhad; the accompanying text identifies the compared measurements as σ/⟨Eν⟩, dσ/dEµ, and dσ/dEhad.Source: Table S1 · p. 14, COMPARISON WITH OUT-OF-TRAINING… · p. 14
• Figure S8 compares base-model and tuned-model predictions with MicroBooNE data for d2σ/dEµd cos θ and dσ/d cos θ.Table S2 gives χ2_c and χ2_v values for the base and tuned models for these distributions.Source: Figure S8 · p. 15, COMPARISON WITH OUT-OF-TRAINING… · p. 15, Table S2 · p. 15

ARGONEUT RESULTS FOR NEUTRINOS IN NEUTRINO MODE.

• Figure S9 presents ArgoNeuT dσ/dθ and dσ/dpµ measurements and base- and tuned-model predictions for neutrinos in the NuMI neutrino beam.The shaded region on the momentum plot marks the pµ range of the MicroBooNE training data.Source: Figure S9 · p. 15, ARGONEUT RESULTS FOR NEUTRINOS IN… · p. 15
• Table S3 compares the MicroBooNE-tuned DNN with ArgoNeuT measurements of muon-neutrino interactions in the NuMI neutrino beam.Source: ARGONEUT RESULTS FOR NEUTRINOS IN… · p. 15, Table S3 · p. 16

ARGONEUT RESULTS FOR NEUTRINOS IN ANTINEUTRINO MODE

• Table S4 reports χ2_v comparisons of the fine-tuned and base models against ArgoNeuT measurements in the antineutrino beam.Source: Table S4 · p. 16, ARGONEUT RESULTS FOR NEUTRINOS IN… · p. 16
• Figure S10 shows ArgoNeuT dσ/dθ and dσ/dpµ measurements with base- and tuned-model predictions for neutrinos in the NuMI antineutrino beam.The shaded regions on the momentum plots mark the pµ range of the MicroBooNE training data.Source: Figure S10 · p. 16, ARGONEUT RESULTS FOR NEUTRINOS IN… · p. 16