Faster Uncertainty Scores for AI Simulations of LHCb’s Cherenkov Detector

Simulating particle collisions and the way detectors respond to them is by far the most computationally expensive task facing the experiments at the Large Hadron Collider. For more than a decade, physicists have explored generative machine learning as a way to replace the interaction-level calculations carried out by the Geant4 libraries with deep neural networks. Neural networks can approximate the outcome of a Geant4 simulation quickly, efficiently, and with hardware acceleration, but they introduce an additional approximation error that must be modeled and controlled. A new study published in EPJ Research Infrastructures addresses exactly that problem for the Ring-Imaging Cherenkov, or RICH, detectors of the LHCb experiment at CERN, proposing a feature-densities method that quantifies uncertainty at a computational cost reduced by an order of magnitude compared with the previously implemented Monte Carlo dropout approach.

The LHCb detector at CERN is a single-arm forward spectrometer that covers heavy flavor, electroweak, and quantum chromodynamics physics. Simulations are vital for understanding and optimizing the detector, but the conventional route relies on Geant4 software to handle the matter–radiation interactions between traversing long-lived particles and the detector. With the luminosity increase expected at the next phase of the LHC, Geant4-based simulations will become too slow to produce the volume of simulations needed for physics analysis and detector performance studies, which is why fast simulation methods are under active investigation.

Earlier work implemented a fully connected Cramér generative adversarial network to simulate the output of the LHCb RICH sub-detector. These detectors are sensitive to particle tracks from kaons, protons, muons, electrons, and particles below the threshold for emitting Cherenkov light. The generative model takes as input the kinematic characteristics of each particle: pseudorapidity, which describes the angle of a particle relative to the beam axis; momentum; the number of tracks detected after the collision; and added random noise. The output consists of five numbers corresponding to the difference between the log-likelihood of each of those particle-type hypotheses and the pion hypothesis, since pions are the most commonly produced particle type at the LHC. These differences are known as RichDLLx values, where x denotes one of the five particle species.

Uncertainty quantification has become a pressing topic as generative models spread across applications, because knowing when to trust a model’s output is essential. In the earlier LHCb work, traditional methods based solely on the model’s output, namely Monte Carlo dropout and Bayesian deep ensembles, were tested. A main drawback of those approaches is the need to compute several model outputs before uncertainty can be quantified. Output-based scoring such as softmax-based methods is also known to suffer from calibration problems, since deep learning models are often over-confident, and no calibration or parameter optimization had been reported in the previous study.

The new method builds on the idea of using embedded representations of the network itself. Feature densities use internal deep learning layers, usually the last one, to project each input into an embedding whose dimensionality is set by the number of neurons in the chosen layer. Using the training data, a histogram is constructed for each embedding feature to approximate a probability density, and the uncertainty score is calculated as one minus the mean of the marginal densities evaluated at the embedding values. This mean assumes independence among the embedding dimensions, a deliberate simplification intended to produce a computationally efficient alternative to ensemble-based methods. For the Cramér GAN architecture used here, every selected layer yields embeddings of dimensionality 128, so 128 densities are estimated. Histograms were chosen because a large training dataset can be used when fitting them, reducing overfitting risk, and because evaluating the histograms at inference time is computationally cheap.

The computational argument is straightforward. Evaluating the feature-densities score costs, per observation, the cost of running the network once plus the cost of evaluating the densities, and the density estimation itself can be done once during training. Monte Carlo dropout, by contrast, requires feeding the same input through the model many times, randomly disabling neurons at each pass, so its cost per observation scales with the number of evaluations. Dropout works by randomly turning neurons on and off during inference, and the resulting spread of outputs serves as the uncertainty value: low dispersion signals low uncertainty, while high dispersion indicates strong disagreement among the model’s predictions and therefore high uncertainty. In the earlier study, Monte Carlo dropout was implemented for the Cramér GAN simulating the RICH detector, making it the natural baseline.

The experiments used a sub-sample of particle identification data collected by LHCb during Run 2, from 2015 to 2018, drawn from pion decay candidates that allow unambiguous identification of one decay product. A quantile transformation with 100,000 quantiles, fitted on a sub-sample of 2,000,000 observations, was applied to the entire dataset before training to approximate a normal distribution, and generated outputs were back-projected to their original units during inference. Following the earlier strategy, the momentum- and pseudorapidity-normalized input space was split along the diagonal line where momentum equals pseudorapidity, producing a training dataset of 948,384 events and a validation dataset of 526,286 events. This splitting strategy was chosen deliberately to create distribution mismatch scenarios between training and validation data.

Three experiments were carried out. The first optimized the most influential parameters: dropout values for Monte Carlo dropout, and embedding layers for feature densities. The best configurations were then compared using the Jensen-Shannon divergence and the Kolmogorov-Smirnov distance between the distributions of generated and real data at each uncertainty band, with the Pearson correlation coefficient between the uncertainty scores and those distances serving as the reliability measure. A well-calibrated generative model should show a high, positive correlation, since there is no single correct output for a given input and standard calibration metrics like the estimated calibration error cannot be applied directly. The results showed that a dropout rate of 0.1 performed best for Monte Carlo dropout, while the third layer of the generator yielded the highest correlation for feature densities. Surprisingly, contrary to the assumption that the final embedding layer is the optimal location for uncertainty assessment, even the initial layer retained sufficient information to produce useful scores.

The comparison confirmed that the feature-densities method predicts uncertainty scores comparable to Monte Carlo dropout. Histograms comparing real and generated RichDLLk values across ten equally sized uncertainty-score bins, each grouping roughly 10,000 observations, showed the same qualitative behavior for both methods, and heatmaps over pseudorapidity, momentum, and number of tracks revealed that the feature-densities method effectively detects regions of low uncertainty at low momentum and pseudorapidity while assigning greater uncertainty to areas with higher momentum and pseudorapidity. An efficiency analysis, which measured how the fraction of events above preset thresholds of 50, 75, and 90 percent deviated between real and generated data, showed that for both methods the higher the uncertainty band, the greater the difference between expected and obtained efficiencies, indicating that the model tends to make more mistakes at higher uncertainty scores.

The decisive factor was speed. Measured on dedicated hardware to eliminate cloud variability, the average inference without uncertainty quantification was 0.006 plus or minus 0.012 seconds per inference. Adding Monte Carlo dropout raised the average computing time to 4.836 plus or minus 0.964 seconds, an increase by a factor of 806 plus or minus 80.33, while feature densities reached 0.354 plus or minus 0.065 seconds, an increase by a factor of 59 plus or minus 5.42. That makes the proposed method around ten times faster than the previously implemented approach. The authors note that a single iteration over the training dataset achieved uncertainty scores commonly exceeding 0.9, though these were sensitive to how probabilities are calculated from the embedding histograms, so calibration of the binning methodology or the density estimation approach, such as kernel density estimation, is needed when applying the method to new datasets. They also suggest future work on quantifying the model’s aleatoric uncertainty, since the generative model reproduces the inherent variability of detector responses but lacks a mechanism to measure it, and on repeatedly generating embeddings from the training data to build more robust histograms.

Subject of Research: Uncertainty quantification for a Cramér GAN-based simulation of the LHCb Ring-Imaging Cherenkov detector using feature densities

Article Title: Uncertainty Quantification for a Cramér Generative Adversarial Network-Based Simulation of the Ring-Imaging Cherenkov Particle Detector Using Feature Densities

Article References: Villalobos-Gómez, E., Ugalde Green, F., Arce Morales, J., Pastor Barrientos, J. D., Anderlini, L., Calderon-Ramirez, S., & Arguedas Cuendis, S. (2026). Uncertainty Quantification for a Cramér Generative Adversarial Network-Based Simulation of the Ring-Imaging Cherenkov Particle Detector Using Feature Densities. EPJ Research Infrastructures, 10(1), Article 15. https://doi.org/10.1007/s41781-026-00166-7

Image Credits: AI Generated

DOI: 10.1007/s41781-026-00166-7

Keywords: uncertainty quantification, Cramér GAN, LHCb, RICH detector, feature densities, Monte Carlo dropout, particle identification, Geant4, generative machine learning, Jensen-Shannon divergence, Kolmogorov-Smirnov distance, computational cost