Neutrino Parameter Estimation from Oscillation Probability Maps

Neutrino oscillations encode fundamental information about neutrino masses and mixing parameters, offering a unique window into physics beyond the Standard Model. Estimating these parameters from oscillation probability maps is, however, computationally challenging due to the maps’ high dimensionality and nonlinear dependence on the underlying physics. In this work, we introduce a data-driven framework that reformulates atmospheric neutrino oscillation parameter inference as a supervised regression task over structured oscillation maps. We propose a hierarchical transformer architecture that explicitly models the two-dimensional structure of these maps, capturing angular dependencies at fixed energies and global correlations across the energy spectrum. To improve physical consistency, the model is trained using a surrogate simulation constraint that enforces agreement between the predicted parameters and the reconstructed oscillation patterns. Furthermore, we introduce a neural network-based uncertainty quantification mechanism that produces distribution-free prediction intervals with formal coverage guarantees.

Overview of the proposed neutrino oscillation parameters estimation.

Installation

The following libraries have to be installed:

Git
Pytorch

To install the package, run pip install -q git+https://github.com/GiorgioMorales/NuOscParam in the terminal.

Download Models

Option 1: Automatic download

After installing the NuOscParam library, run the following script on Python (or on the terminal without the "!"):

!hf download GiorgioMorales/NuInferenceModels \
  --repo-type model \
  --local-dir $(python -c "import NuOscParam; import os; print(os.path.join(os.path.dirname(NuOscParam.__file__), 'Models', 'saved_models'))") \
  --include "ModelType-HierarchicalTransformer/*"

Option 2: Manual Download

If, instead, you cloned the repository, download the model manually from HugginFace. Then, copy and paste them into the src/NuOscParam/Models/saved_models/ModelType-HierarchicalTransformer folder.

Download Datasets

In case you need to retrain the models or simply access the datasets, download them from HugginFace. Then, copy and paste them into the src/NuOscParam/Models/saved_models/ModelType-HierarchicalTransformer folder.

Usage

Generate Neutrino Oscillation Maps

We use a simulator that generates 9 oscillation probability maps:

Transition ↓ / Source →	$\nu_e$ (source)	$\nu_\mu$ (source)	$\nu_\tau$ (source)
$\nu_e$ (detected)	$P(\nu_e \leftarrow \nu_e)$	$P(\nu_e \leftarrow \nu_\mu)$	$P(\nu_e \leftarrow \nu_\tau)$
$\nu_\mu$ (detected)	$P(\nu_\mu \leftarrow \nu_e)$	$P(\nu_\mu \leftarrow \nu_\mu)$	$P(\nu_\mu \leftarrow \nu_\tau)$
$\nu_\tau$ (detected)	$P(\nu_\tau \leftarrow \nu_e)$	$P(\nu_\tau \leftarrow \nu_\mu)$	$P(\nu_\tau \leftarrow \nu_\tau)$

In all cases, the $\nu$ oscillation parameters follow the order: [theta12, theta23, theta13, delta_cp, m21, m31].

Generate Maps Using Known Oscillation Parameters

To use a exact simulator, initiate the Simulator class with the following parameters:

Parameters:

mode: It can take the values vacuum, which will use a simulator that produces $\nu$ oscillation maps in vacuum, or earth (default), which will use a simulator that produces $\nu$ oscillation maps after Earth-matter effect.
cropR: It determines the number of rows ($\theta$) bins are used. Min:1, Max:120, Default: 80.
cropC: It determines the number of columns (energy) bins are used. Min:1, Max:120, Default: 30.

from NuOscParam.utils import *
from NuOscParam.Data.DataRanges import *
from NuOscParam.Data.Simulator import Simulator
generator = Simulator(ranges=NEUTRINO_RANGES, mode='earth', device=torch.device("cuda" if torch.cuda.is_available() else "cpu"))

To get the actual maps given a set of $\nu$ oscillation parameters, we call the get_maps method. The following example considers two sets of 6 $\nu$ oscillation parameters:

import numpy as np

batch = np.array([[3.2434875e+01, 5.0723694e+01, 8.4253168e+00, 1.8559186e+02, 6.9126829e-05, 2.4357950e-03],
              [33.003803, 51.50117, 8.243288, 243.95, 7.993506e-05, 0.0024312385]])
maps = generator.get_maps(batch)
# Optionally, plot maps
for i in range(len(batch)):
    plot_osc_maps(maps[i, :, :, :].permute(1, 2, 0), title=f"Oscillation Maps. Sample {i+1}")

Generate Random Maps

To generate random $\nu$ oscillation maps and corresponding oscillation parameters, we use the OscIterableDataset iterable class. Its parameters are the same as those from the Simulator class:

generator = iter(OscIterableDataset(ranges=NEUTRINO_RANGES, mode='earth',
                                    device=torch.device("cuda" if torch.cuda.is_available() else "cpu")))

To generate random maps, we call the next command iterably:

X_test, Osc_params = [], []
n_samples = 3  # Generate three random sets of maps
for i in range(n_samples):
    xtest, _, osc_pars, _ = next(generator)  # Generate 1 sample
    X_test.append(xtest)
    Osc_params.append(osc_pars)
    # Plot oscillation maps    
    plot_osc_maps(input_image = xtest[0, :, :, :].permute(1, 2, 0), title=f"Oscillation Maps. Sample {i+1}")
X_test = torch.cat(X_test, dim=0)

Proposed Neutrino Parameter Estimation from Oscillation Probability Maps

To use our $\nu$ oscillation parameter inference framework based on the use of Structured Hierarchical Transformers, call the OscillationEstimator class:

from NuOscParam.OscillationEstimator import OscillationEstimator
predictor = OscillationEstimator()

We use this class and the predict method to process a batch of $\nu$ oscillation maps:

Parameters:

denormalize: If True, the resulting oscillation parameters are given in their normal (operating) scale. If False (default), they're still normalized (between 0 and 1).
uncertainty: If False, only the point predictions are retrieved using the trained transformers. If True, the Conformal DualAQD is used in addition to produce 90% prediction intervals. See the Demo for more details.

pred_OscParams = predictor.predict(X_test, denormalize=True, uncertainty=True)