Neural network approximation of Euclidean path integrals and its application for the $φ^4$ theory in 1+1 dimensions

Abstract: Studying phase transitions in interacting quantum field theories generally requires the numerical study of the dynamical system on an N-dimensional lattice, which is, in most cases, computationally quite the challenging task even with modern computing facilities. In this work I propose an alternative way to solve Euclidean path integrals in quantum field theories, using radial basis function-type neural networks, where the nonlinear part of the path integral is approximated by a linear combination of quadratic path integrals, therefore making the corresponding problem analytically tractable. The method allows us to calculate observables in a very efficient way, taking only seconds to do calculations that would otherwise take hours or even days with other existing methods. To test the capabilities of the model, it is used to describe the phase transition in the 1+1 dimensional interacting real scalar $\phi^4$ theory. The obtained phase transition line is compared to previous lattice results, giving very good agreement between them. The method is very flexible and could be extended to higher dimensions, finite temperatures, or even finite densities, therefore, it could give a very good alternative approach for solving numerically hard problems in quantum field theories.