Neural network field theory formulates field theory as a statistical ensemble of fields defined by a network architecture and a density on its parameters. We extend the construction to topological settings via the inclusion of discrete parameters that label the topological quantum number. We recover the Berezinskii--Kosterlitz--Thouless transition, including the spin-wave critical line and the proliferation of vortices at high temperatures. We also verify the T-duality of the bosonic string, showing invariance under the exchange of momentum and winding on $S^1$, the transformation of the sigma model couplings according to the Buscher rules on constant toroidal backgrounds, the enhancement of the current algebra at self-dual radius, and non-geometric T-fold transition functions.

翻译：神经网络场论将场论构建为由网络架构及其参数上的密度所定义的场的统计系综。我们通过引入标记拓扑量子数的离散参数，将该构造推广至拓扑设置。我们恢复了别列津斯基-科斯特利茨-索利斯相变，包括自旋波临界线以及高温下涡旋的增殖。我们还验证了玻色弦的T对偶性，证明了在$S^1$上动量与缠绕模式的交换不变性、根据布谢尔规则在常数环面背景下的西格玛模型耦合变换、自对偶半径处电流代数的增强，以及非几何T-折叠转移函数。