# Reconstructing S-matrix Phases with Machine Learning

arXiv: https://arxiv.org/abs/2308.09451 · HTML: https://mlsanity.com/paper/2308.09451/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 56 references that are in the map.
- First posted on arXiv 18 Aug 2023.

## Builds on (most cited first, 7 of 7)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [AI Feynman: a Physics-Inspired Method for Symbolic Regression](https://mlsanity.com/paper/1905.11481/) (2019)
- [CaloGAN: Simulating 3D High Energy Particle Showers in Multi-Layer Electromagnetic Calorimeters with Generative Adversarial Networks](https://mlsanity.com/paper/1712.10321/) (2017)
- [Deep Generative Symbolic Regression with Monte-Carlo-Tree-Search](https://mlsanity.com/paper/2302.11223/) (2023)
- [nPINNs: nonlocal Physics-Informed Neural Networks for a parametrized nonlocal universal Laplacian operator. Algorithms and Applications](https://mlsanity.com/paper/2004.04276/) (2020)

## Built on by (most cited first, 4 of 4)

- [Truth, beauty, and goodness in grand unification: a machine learning approach](https://mlsanity.com/paper/2411.06718/) (2024)
- [Transforming Calabi-Yau Constructions: Generating New Calabi-Yau Manifolds with Transformers](https://mlsanity.com/paper/2507.03732/) (2025)
- [Learning S-Matrix Phases with Neural Operators](https://mlsanity.com/paper/2404.14551/) (2024)
- [Deep Neural Network Driven Simulation Based Inference Method for Pole Position Estimation under Model Misspecification](https://mlsanity.com/paper/2507.18824/) (2025)

## Abstract (arXiv)

An important element of the S-matrix bootstrap program is the relationship between the modulus of an S-matrix element and its phase. Unitarity relates them by an integral equation. Even in the simplest case of elastic scattering, this integral equation cannot be solved analytically and numerical approaches are required. We apply modern machine learning techniques to studying the unitarity constraint. We find that for a given modulus, when a phase exists it can generally be reconstructed to good accuracy with machine learning. Moreover, the loss of the reconstruction algorithm provides a good proxy for whether a given modulus can be consistent with unitarity at all. In addition, we study the question of whether multiple phases can be consistent with a single modulus, finding novel phase-ambiguous solutions. In particular, we find a new phase-ambiguous solution which pushes the known limit on such solutions significantly beyond the previous bound.