Addressing the sign problem in Euclidean path integrals with radial basis function neural networks

Solving interacting field theories at finite densities remains a numerically and conceptually challenging task, even with modern computational capabilities. In this paper, we propose a novel approach based on an expansion of the Euclidean path integrals using radial basis function neural networks, which allows the calculation of observables at finite densities and overcomes the sign problem in a numerically very efficient manner. The method is applied to an interacting complex scalar field theory at finite chemical potential in <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mrow> <a:mn>3</a:mn> <a:mo>+</a:mo> <a:mn>1</a:mn> </a:mrow> </a:math> dimensions, which exhibits both the sign problem and the silver blaze phenomenon, similar to QCD. The critical chemical potential at which phase transition occurs is estimated to be <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:msub> <c:mi>μ</c:mi> <c:mi>c</c:mi> </c:msub> <c:mo>=</c:mo> <c:mn>1.17</c:mn> <c:mo>±</c:mo> <c:mn>0.018</c:mn> </c:math> , and the silver blaze problem is accurately described below <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:msub> <e:mi>μ</e:mi> <e:mi>c</e:mi> </e:msub> </e:math> .