Gauge covariant neural network for 4 dimensional non-abelian gauge theory

Abstract

We develop a gauge covariant neural network for four dimensional non-abelian gauge theory, which realizes a map between rank-2 tensor valued vector fields.We find that the conventional smearing procedure and gradient flow for gauge fields can be regarded as known neural networks, residual networks and neural ordinal differential equations for rank-2 tensors with fixed parameters.In terms of machine learning context, projection or normalization functions in the smearing schemes correspond to an activation function in neural networks.Using the locality of the activation function, we derive the backpropagation for the gauge covariant neural network. Consequently, the smeared force in hybrid Monte Carlo (HMC) is naturally derived with the backpropagation. As a demonstration, we develop the self-learning HMC (SLHMC) with covariant neural network approximated action for non-abelian gauge theory with dynamical fermions, and we observe SLHMC reproduces results from HMC.

Gauge invariant input to neural network for path optimization method

“…One possible direction is the incorporation of larger Wilson loops and the Polyakov lines as input to the neural network. Another direction is the adoption of novel approaches which respect the gauge symmetry in the neural network, such as the lattice gauge equivariant convolutional neural networks [36] and the gauge covariant neural network [46]. In addition, there is a different approach that modifies the action instead of the integral path [47][48][49].…”

Efficient Modelling of Trivializing Maps for Lattice $ϕ^4$ Theory Using Normalizing Flows: A First Look at Scalability

“…A more general convolutional architecture was introduced in Reference [98], which allows for equivariance under a wider class of global symmetries, including rotations and discrete transformations. Building on these foundations, more recent developments include convolutional networks that are equivariant under Lie groups [99][100][101][102]. 15 There is no reason why the couplings of the theory have to remain constant during training.…”

Flow-based sampling for fermionic lattice field theories

“…The target densities defined in Table I are all invariant under translations with appropriate boundary conditions, as discussed in Section II C. Previous works have shown that exactly incorporating known symmetries into machine learning models can accelerate their training and improve their final quality [86][87][88][89][90][91]. In the context of normalizing flows, ensuring that the model density is invariant under a symmetry group is achieved by choosing an invariant prior distribution and building transformation layers that are equivariant under the symmetry.…”