# SEAL - A Symmetry EncourAging Loss for High Energy Physics

arXiv: https://arxiv.org/abs/2511.01982 · HTML: https://mlsanity.com/paper/2511.01982/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 24 references that are in the map.
- First posted on arXiv 3 Nov 2025.

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

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Particle Transformer for Jet Tagging](https://mlsanity.com/paper/2202.03772/) (2022)
- [Lorentz-Equivariant Geometric Algebra Transformers for High-Energy Physics](https://mlsanity.com/paper/2405.14806/) (2024)
- [Equivariance versus Augmentation for Spherical Images](https://mlsanity.com/paper/2202.03990/) (2022)
- [Explainable Equivariant Neural Networks for Particle Physics: PELICAN](https://mlsanity.com/paper/2307.16506/) (2023)
- [Lie Point Symmetry and Physics Informed Networks](https://mlsanity.com/paper/2311.04293/) (2023)
- [A General Framework for Equivariant Neural Networks on Reductive Lie Groups](https://mlsanity.com/paper/2306.00091/) (2023)
- [Revisiting Data Augmentation for Rotational Invariance in Convolutional Neural Networks](https://mlsanity.com/paper/2310.08429/) (2023)
- [Long Short-Term Memory (LSTM) networks with jet constituents for boosted top tagging at the LHC](https://mlsanity.com/paper/1711.09059/) (2017)
- [Relaxed Equivariance via Multitask Learning](https://mlsanity.com/paper/2410.17878/) (2024)
- [Semi-Equivariant GNN Architectures for Jet Tagging](https://mlsanity.com/paper/2202.06941/) (2022)
- [Learning Broken Symmetries with Approximate Invariance](https://mlsanity.com/paper/2412.18773/) (2024)

## Built on by (most cited first, 1 of 1)

- [Generative models on phase space](https://mlsanity.com/paper/2604.02415/) (2026)

## Abstract (arXiv)

Physical symmetries provide a strong inductive bias for constructing functions to analyze data. In particular, this bias may improve robustness, data efficiency, and interpretability of machine learning models. However, building machine learning models that explicitly respect symmetries can be difficult due to the dedicated components required. Moreover, real-world experiments may not exactly respect fundamental symmetries at the level of finite granularities and energy thresholds. In this work, we explore an alternative approach to create symmetry-aware machine learning models. We introduce soft constraints that allow the model to decide the importance of added symmetries during the learning process instead of enforcing exact symmetries. We investigate two complementary approaches, one that penalizes the model based on specific transformations of the inputs and one inspired by group theory and infinitesimal transformations of the inputs. Using top quark jet tagging and Lorentz equivariance as examples, we observe that the addition of the soft constraints leads to more robust performance while requiring negligible changes to current state-of-the-art models.