Learning S-Matrix Phases with Neural Operators

, , , , 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.

Introduction

The study of scattering amplitudes in theoretical physics has been a topic of interest for decades. These amplitudes, which describe the probability of particles interacting and exchanging energy during collisions, are crucial in understanding the fundamental laws of nature. In particular, the relationship between the modulus and phase of these amplitudes is essential in S-matrix theory, where unitarity constraints govern this connection through a complex integral equation.

However, solving this equation directly can be challenging due to its complexity. To address this issue, researchers have turned to data-driven techniques like Machine Learning to uncover hidden structures and relationships within scattering amplitudes. In this research paper, titled "Learning Modulus-Phase Relations from Data with Fourier Neural Operators," authors Jiequn Han and Yann LeCun propose a novel approach using Fourier Neural Operators (FNOs) to learn and predict the phases of scattering amplitudes from data.

The Methodology

The methodology employed by Han and LeCun involves training FNOs on a dataset comprising both true and false samples of scattering amplitudes. The goal is to accurately predict the phases even when dealing with infinite partial wave expansions. This presents an unconventional application for FNOs as it involves a simultaneous regression-classification task.

To train their model, the authors use a dataset consisting of 10 million random samples generated from known amplitude profiles with varying degrees of noise added. They also include some false samples that violate unitarity constraints to test their model's ability to distinguish between allowed and disallowed profiles.

FNOs: A Brief Overview

Fourier Neural Operators (FNOs) are deep neural networks designed specifically for learning function mappings between high-dimensional spaces efficiently. They leverage ideas from classical harmonic analysis such as Fourier transforms to approximate functions using sinusoidal basis functions instead of traditional polynomial ones.

The Results

The results of this study are impressive, with the FNOs accurately predicting the phases of scattering amplitudes even when dealing with infinite partial wave expansions. Additionally, the authors observe a strong correlation between their model's prediction accuracy and the violation of unitarity constraints for the predicted phases. This highlights the efficacy of their approach in distinguishing between allowed and disallowed profiles of the modulus.

Implications and Future Directions

This research has significant implications for both Physics and Mathematics. By leveraging data-driven techniques like Machine Learning, Han and LeCun have demonstrated how modern approaches can offer new perspectives on longstanding problems in these fields. In particular, their methodology could be applied to non-perturbative Quantum Field Theories (QFTs), where traditional methods often fall short due to computational challenges.

Moreover, this study also sheds light on the potential limitations of using FNOs in theoretical physics. While they prove effective in learning function mappings from data, it is essential to consider statistical factors when dealing with ensembles of Neural Operators (NOs). Further research could explore ways to address these limitations and improve upon existing methodologies.

Conclusion

In conclusion, "Learning Modulus-Phase Relations from Data with Fourier Neural Operators" presents a novel approach for studying complex function mappings in theoretical physics through data-driven insights. By training FNOs on a dataset comprising both true and false samples of scattering amplitudes, Han and LeCun demonstrate how modern techniques can offer new perspectives on longstanding problems in Physics and Mathematics.

Their findings not only contribute to advancing our understanding of function relationships but also pave the way for innovative methodologies that combine data-driven insights with analytical approaches. Through their exploration of scattering amplitudes using FNOs, this research showcases how collaborations between different disciplines can lead to groundbreaking discoveries in science.