# Solving differential equations with neural networks: Applications to the calculation of cosmological phase transitions

## TL;DR

This paper introduces a neural network-based method for solving differential equations, demonstrating its effectiveness in calculating cosmological phase transition profiles with accuracy comparable or superior to traditional solvers.

## Contribution

The authors present a novel neural network approach for solving differential equations that is flexible, stable, and applicable to various types, outperforming existing specialized numerical solvers in some cases.

## Key findings

- Neural network method achieves at least as accurate results as dedicated solvers.
- The approach is applicable to ordinary, partial, and coupled differential equations.
- Neural networks can find solutions where traditional solvers fail.

## Abstract

Starting from the observation that artificial neural networks are uniquely suited to solving optimisation problems, and most physics problems can be cast as an optimisation task, we introduce a novel way of finding a numerical solution to wide classes of differential equations. We find our approach to be very flexible and stable without relying on trial solutions, and applicable to ordinary, partial and coupled differential equations. We apply our method to the calculation of tunnelling profiles for cosmological phase transitions, which is a problem of relevance for baryogenesis and stochastic gravitational wave spectra. Comparing our solutions with publicly available codes which use numerical methods optimised for the calculation of tunnelling profiles, we find our approach to provide at least as accurate results as these dedicated differential equation solvers, and for some parameter choices even more accurate and reliable solutions. In particular, we compare the neural network approach with two publicly available profile solvers, \texttt{CosmoTransitions} and \texttt{BubbleProfiler}, and give explicit examples where the neural network approach finds the correct solution while dedicated solvers do not. We point out that this approach of using artificial neural networks to solve equations is viable for any problem that can be cast into the form $\mathcal{F}(\vec{x})=0$, and is thus applicable to various other problems in perturbative and non-perturbative quantum field theory.