# Learning S-Matrix Phases with Neural Operators

**Authors:** V. Niarchos, C. Papageorgakis  
**arXiv:** [2404.14551](https://arxiv.org/abs/2404.14551)  
**Date:** 2024-04-22  
**Summary page:** https://www.summarizepaper.com/en/paper/2404.14551v1/

## Key points

- Investigating relationship between modulus and phase of scattering amplitudes in elastic 2 → 2 scattering at fixed energies
- Using Fourier Neural Operators (FNOs) to learn relationship from data on amplitudes with finite partial wave expansions
- FNOs accurately predict phases even with infinite partial wave expansions, providing a classification index for prediction quality
- Strong correlation between classification index and violation of unitarity constraints for predicted phases
- Unconventional application of FNOs involves regression-classification task, emphasizing importance of statistics in ensembles of Neural Operators (NOs)
- Potential of FNOs as new methodology for studying complex function mappings in Theoretical Physics
- Contribution to advancing understanding of function relationships in theoretical frameworks and bridging gaps between explicit solutions and elusive structures using data-driven techniques like Machine Learning

## Summary

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In this study, we investigate the relationship between the modulus and phase of scattering amplitudes in elastic 2 → 2 scattering at fixed energies. This connection is crucial in S-matrix theory and is typically governed by unitarity constraints through a complex integral equation. However, instead of directly solving this equation, we employ a novel approach using Fourier Neural Operators (FNOs) to learn and uncover this relationship from data on amplitudes with finite partial wave expansions. Our methodology involves training FNOs on a dataset comprising both true and false samples of amplitudes, aiming to accurately predict the phases even when dealing with infinite partial wave expansions. Remarkably, our results show that the FNOs not only predict these phases but also provide a classification index indicating the quality of their predictions. We observe a strong correlation between this index and the violation of unitarity constraints for the predicted phases, highlighting the efficacy of our approach in distinguishing between allowed and disallowed profiles of the modulus. This unconventional application of FNOs presents a unique challenge as it involves a simultaneous regression-classification task, emphasizing the importance of statistics in ensembles of Neural Operators (NOs). Our findings shed light on both the merits and limitations of using FNOs in Theoretical Physics, showcasing its potential as a new methodology for studying complex function mappings in various contexts. Furthermore, we contextualize our research within the broader landscape of Physics and Mathematics where understanding maps between functions is paramount. Particularly in non-perturbative Quantum Field Theories (QFTs), where traditional methods often fall short, novel approaches like ours hold promise for uncovering hidden structures and generalizing towards computationally challenging scenarios. By leveraging data-driven techniques like Machine Learning, we aim to bridge gaps between explicit solutions in special cases and elusive structures in generic regimes. Overall, our study contributes to advancing our understanding of function relationships in theoretical frameworks while paving the way for innovative methodologies that combine data-driven insights with analytical approaches. Through our exploration of scattering amplitudes using FNOs, we demonstrate how modern techniques can offer new perspectives on longstanding problems in Physics and Mathematics.