# Flexible Spectral-Normalized Neural Gaussian Process for Dynamic Aperture Prediction

arXiv: https://arxiv.org/abs/2609.08620 · HTML: https://mlsanity.com/paper/2609.08620/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 30 references that are in the map.
- First posted on arXiv 8 Sep 2026.

## Builds on (most cited first, 13 of 13)

- [On Calibration of Modern Neural Networks](https://mlsanity.com/paper/1706.04599/) (2017)
- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [Deep Deterministic Uncertainty: A Simple Baseline](https://mlsanity.com/paper/2102.11582/) (2021)
- [Quantifying Aleatoric and Epistemic Uncertainty in Machine Learning: Are Conditional Entropy and Mutual Information Appropriate Measures?](https://mlsanity.com/paper/2209.03302/) (2022)
- [On the Practicality of Deterministic Epistemic Uncertainty](https://mlsanity.com/paper/2107.00649/) (2021)
- [A Simple Approach to Improve Single-Model Deep Uncertainty via Distance-Awareness](https://mlsanity.com/paper/2205.00403/) (2022)
- [Rethinking Aleatoric and Epistemic Uncertainty](https://mlsanity.com/paper/2412.20892/) (2024)
- [Autonomous Driving with Deep Learning: A Survey of State-of-Art Technologies](https://mlsanity.com/paper/2006.06091/) (2020)
- [On Second-Order Scoring Rules for Epistemic Uncertainty Quantification](https://mlsanity.com/paper/2301.12736/) (2023)
- [Correlated Input-Dependent Label Noise in Large-Scale Image Classification](https://mlsanity.com/paper/2105.10305/) (2021)
- [Introducing an Improved Information-Theoretic Measure of Predictive Uncertainty](https://mlsanity.com/paper/2311.08309/) (2023)
- [Deep Classifiers with Label Noise Modeling and Distance Awareness](https://mlsanity.com/paper/2110.02609/) (2021)

## Abstract (arXiv)

We address the challenge of scalable uncertainty quantification in large-scale scientific applications, where complex state-of-the-art machine learning methods are often computationally infeasible. Our primary contribution is a simple yet effective empirical Bayes method for automatically tuning the hyperparameters of a flexible, heteroscedastic Spectral-normalized Neural Gaussian Process. This approach retains the expressiveness and uncertainty-awareness of semi-Bayesian neural models while significantly reducing the computational burden by integrating hyperparameter learning directly into the training loop. We demonstrate the practical impact of our method on the task of estimating the dynamic aperture in circular particle accelerators, a fundamental problem in high-energy physics colliders and storage rings, using simulation data from the case of the Large Hadron Collider at CERN. Traditional approaches to DA estimation require extensive particle-tracking simulations, which are prohibitively time-consuming and resource-intensive. Our results show that the proposed method achieves competitive predictive performance and well-calibrated uncertainty estimates at much lower computational cost than state-of-the-art approaches. We stress that, beyond this application, the proposed empirical Bayes framework offers a general solution for training heteroscedastic neural models in situations where manual hyperparameter tuning is impractical. Accordingly, we anticipate that this framework can be applied to other domains that encounter comparable computational limitations.