# Learning S-Matrix Phases with Neural Operators

arXiv: https://arxiv.org/abs/2404.14551 · HTML: https://mlsanity.com/paper/2404.14551/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 36 references that are in the map.
- First posted on arXiv 22 Apr 2024.

## Builds on (most cited first, 8 of 8)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- ["Zero-Shot" Super-Resolution using Deep Internal Learning](https://mlsanity.com/paper/1712.06087/) (2017)
- [Scattering with Neural Operators](https://mlsanity.com/paper/2308.14789/) (2023)
- [Reconstructing S-matrix Phases with Machine Learning](https://mlsanity.com/paper/2308.09451/) (2023)

## Built on by (most cited first, 3 of 3)

- [Bayesian RG Flow in Neural Network Field Theories](https://mlsanity.com/paper/2405.17538/) (2024)
- [Neural Quantum Propagators for Driven-Dissipative Quantum Dynamics](https://mlsanity.com/paper/2410.16091/) (2024)
- [Operator Learning for Schr"odinger Equation: Unitarity, Error Bounds, and Time Generalization](https://mlsanity.com/paper/2505.18288/) (2025)

## Abstract (arXiv)

We use Fourier Neural Operators (FNOs) to study the relation between the modulus and phase of amplitudes in 2→ 2 elastic scattering at fixed energies. Unlike previous approaches, we do not employ the integral relation imposed by unitarity, but instead train FNOs to discover it from many samples of amplitudes with finite partial wave expansions. When trained only on true samples, the FNO correctly predicts (unique or ambiguous) phases of amplitudes with infinite partial wave expansions. When also trained on false samples, it can rate the quality of its prediction by producing a true/false classifying index. We observe that the value of this index is strongly correlated with the violation of the unitarity constraint for the predicted phase, and present examples where it delineates the boundary between allowed and disallowed profiles of the modulus. Our application of FNOs is unconventional: it involves a simultaneous regression-classification task and emphasizes the role of statistics in ensembles of NOs. We comment on the merits and limitations of the approach and its potential as a new methodology in Theoretical Physics.