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.
Highlights
A neural network is an algorithm designed to perform an optimization procedure, where the loss function provides a measure of the performance of the optimization
If a physics problem can be cast into the form F ðx⃗ Þ 1⁄4 0, its solution can be calculated by minimizing the loss function of a neural network
By building on the capabilities of an artificial neural network in solving optimization problems, we have proposed a novel way to find solutions to differential equations
Summary
A neural network is an algorithm designed to perform an optimization procedure, where the loss function provides a measure of the performance of the optimization. If a physics problem can be cast into the form F ðx⃗ Þ 1⁄4 0, its solution can be calculated by minimizing the loss function of a neural network While this approach is applicable to any function F , we attempt to apply this observation to the solution of differential equations and to the nonperturbative calculation of tunneling rates of electroweak phase transitions. We propose to use these powerful artificial neural network algorithms in a different way, namely to directly find solutions to differential equations. We apply these methods to calculate the solution of the nonperturbative quantum-fieldtheoretical description of tunneling processes for electroweak phase transitions. We will begin by describing the method in detail and showcasing how it can be used to solve differential equations of varying complexity, before applying it to the calculation of cosmological phase transitions
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