# Reconstructing spectral functions via automatic differentiation

arXiv: https://arxiv.org/abs/2111.14760 · HTML: https://mlsanity.com/paper/2111.14760/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 1 of its 43 references that are in the map.
- First posted on arXiv 29 Nov 2021.

## Builds on (most cited first, 1 of 1)

- [NETT: Solving Inverse Problems with Deep Neural Networks](https://mlsanity.com/paper/1803.00092/) (2018)

## Built on by (most cited first, 5 of 5)

- [Physics-Driven Learning for Inverse Problems in Quantum Chromodynamics](https://mlsanity.com/paper/2501.05580/) (2025)
- [Automatic differentiation approach for reconstructing spectral functions with neural networks](https://mlsanity.com/paper/2112.06206/) (2021)
- [Machine learning Hadron Spectral Functions in Lattice QCD](https://mlsanity.com/paper/2112.00460/) (2021)
- [Neural Networks Asymptotic Behaviours for the Resolution of Inverse Problems](https://mlsanity.com/paper/2402.09338/) (2024)
- [Machine Learning Meets High-Energy Nuclear Physics: From Pattern Recognition to Physics-Integrated Discovery](https://mlsanity.com/paper/2610.12293/) (2026)

## Abstract (arXiv)

Reconstructing spectral functions from Euclidean Green's functions is an important inverse problem in many-body physics. However, the inversion is proved to be ill-posed in the realistic systems with noisy Green's functions. In this Letter, we propose an automatic differentiation(AD) framework as a generic tool for the spectral reconstruction from propagator observable. Exploiting the neural networks' regularization as a non-local smoothness regulator of the spectral function, we represent spectral functions by neural networks and use the propagator's reconstruction error to optimize the network parameters unsupervisedly. In the training process, except for the positive-definite form for the spectral function, there are no other explicit physical priors embedded into the neural networks. The reconstruction performance is assessed through relative entropy and mean square error for two different network representations. Compared to the maximum entropy method, the AD framework achieves better performance in the large-noise situation. It is noted that the freedom of introducing non-local regularization is an inherent advantage of the present framework and may lead to substantial improvements in solving inverse problems.