# Using Symbolic Regression to Emulate the Radial Fourier Transform of the S\'ersic profile for Fast, Accurate and Differentiable Galaxy Profile Fitting

## TL;DR

This paper develops a symbolic regression-based emulator for the Fourier transform of Sersic galaxy profiles, enabling faster and differentiable galaxy profile fitting with minimal accuracy loss.

## Contribution

The authors create a computationally efficient, differentiable emulator for the Sersic profile's Fourier transform using symbolic regression, improving fitting speed and scalability.

## Key findings

- Galaxy profile fitting is accelerated by 2.5 times using the emulator.
- The Fourier transform varies smoothly with Sersic index and frequency.
- The emulator maintains accuracy in both simulated and real galaxy data.

## Abstract

Galaxy profile fitting is a ubiquitous technique that provides the backbone for photometric and morphological measurements in modern extragalactic surveys. A recent innovation in profile fitting algorithms is to render, or create, the model profile in Fourier space, which aims to provide faster and more accurate results. However, the most common parameterization, the S\'ersic profile, has no closed form Fourier transform, requiring the use of computationally expensive approximations. In this paper our goal is to develop an emulator to mimic the radial Fourier transform of the S\'ersic profile, for use in profile fitting. We first numerically compute the radial Fourier transform and demonstrate that it varies smoothly as a function of the S\'ersic index and $k$, the spatial frequency coordinate. Using this set of numerically calculated transforms as a training set, we use symbolic regression to discover an equation which approximates its behavior. This ensures the emulator will be based on computationally efficient and differentiable building blocks. We implement this novel rendering method in the pysersic profile fitter, and ensure it is accurate by conducting both injection-recovery tests using model galaxy profiles and applying multiple rendering methods to a real sample of galaxies in HSC-SSP imaging. Crucially, the Fourier emulator rendering technique enables measurements of morphological parameters of galaxies 2.5 times faster than standard methods with minimal loss in accuracy. This increased performance while maintaining accuracy is a step that ensures these tools can continue to scale with the ever-increasing flow of incoming data.